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</div> </a> <ul id="toc-Regular_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Morphism_of_affine_varieties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Morphism_of_affine_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>Morphism of affine varieties</span> </div> </a> <ul id="toc-Morphism_of_affine_varieties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rational_function_and_birational_equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_function_and_birational_equivalence"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Rational function and birational equivalence</span> </div> </a> <ul id="toc-Rational_function_and_birational_equivalence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_variety" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_variety"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>Projective variety</span> </div> </a> <ul id="toc-Projective_variety-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Real_algebraic_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Real_algebraic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Real algebraic geometry</span> </div> </a> <ul id="toc-Real_algebraic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computational_algebraic_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computational_algebraic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Computational algebraic geometry</span> </div> </a> <button aria-controls="toc-Computational_algebraic_geometry-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Computational algebraic geometry subsection</span> </button> <ul id="toc-Computational_algebraic_geometry-sublist" class="vector-toc-list"> <li id="toc-Gröbner_basis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gröbner_basis"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Gröbner basis</span> </div> </a> <ul id="toc-Gröbner_basis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cylindrical_algebraic_decomposition_(CAD)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cylindrical_algebraic_decomposition_(CAD)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Cylindrical algebraic decomposition (CAD)</span> </div> </a> <ul id="toc-Cylindrical_algebraic_decomposition_(CAD)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Asymptotic_complexity_vs._practical_efficiency" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Asymptotic_complexity_vs._practical_efficiency"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Asymptotic complexity vs. practical efficiency</span> </div> </a> <ul id="toc-Asymptotic_complexity_vs._practical_efficiency-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Abstract_modern_viewpoint" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Abstract_modern_viewpoint"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Abstract modern viewpoint</span> </div> </a> <ul id="toc-Abstract_modern_viewpoint-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle History subsection</span> </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Before_the_16th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Before_the_16th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Before the 16th century</span> </div> </a> <ul id="toc-Before_the_16th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Renaissance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Renaissance"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Renaissance</span> </div> </a> <ul id="toc-Renaissance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-19th_and_early_20th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#19th_and_early_20th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>19th and early 20th century</span> </div> </a> <ul id="toc-19th_and_early_20th_century-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-20th_century" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#20th_century"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>20th century</span> </div> </a> <ul id="toc-20th_century-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Analytic_geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Analytic_geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Analytic geometry</span> </div> </a> <ul id="toc-Analytic_geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algebraic geometry</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 62 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-62" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">62 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D8%A9_%D8%AC%D8%A8%D8%B1%D9%8A%D8%A9" title="هندسة جبرية – Arabic" lang="ar" hreflang="ar" data-title="هندسة جبرية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Xeometr%C3%ADa_alxebraica" title="Xeometría alxebraica – Asturian" lang="ast" hreflang="ast" data-title="Xeometría alxebraica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%80%E0%A6%9C%E0%A6%97%E0%A6%BE%E0%A6%A3%E0%A6%BF%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%9C%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AE%E0%A6%BF%E0%A6%A4%E0%A6%BF" title="বীজগাণিতিক জ্যামিতি – Bangla" lang="bn" hreflang="bn" data-title="বীজগাণিতিক জ্যামিতি" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебраик геометрия – Bashkir" lang="ba" hreflang="ba" data-title="Алгебраик геометрия" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D0%B5%D0%B0%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%8F" title="Алгебраічная геаметрыя – Belarusian" lang="be" hreflang="be" data-title="Алгебраічная геаметрыя" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%90%D0%BB%D1%8C%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D1%96%D1%87%D0%BD%D0%B0%D1%8F_%D0%B3%D0%B5%D0%B0%D0%BC%D1%8D%D1%82%D1%80%D1%8B%D1%8F" title="Альгебраічная геамэтрыя – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Альгебраічная геамэтрыя" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебрична геометрия – Bulgarian" lang="bg" hreflang="bg" data-title="Алгебрична геометрия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-br mw-list-item"><a href="https://br.wikipedia.org/wiki/Mentoniezh_aljebrek" title="Mentoniezh aljebrek – Breton" lang="br" hreflang="br" data-title="Mentoniezh aljebrek" data-language-autonym="Brezhoneg" data-language-local-name="Breton" class="interlanguage-link-target"><span>Brezhoneg</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Geometria_algebraica" title="Geometria algebraica – Catalan" lang="ca" hreflang="ca" data-title="Geometria algebraica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%C4%83%D0%BB%D0%BB%C4%83_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8" title="Алгебрăллă геометри – Chuvash" lang="cv" hreflang="cv" data-title="Алгебрăллă геометри" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algebraick%C3%A1_geometrie" title="Algebraická geometrie – Czech" lang="cs" hreflang="cs" data-title="Algebraická geometrie" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Geometreg_algebraidd" title="Geometreg algebraidd – Welsh" lang="cy" hreflang="cy" data-title="Geometreg algebraidd" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebraische_Geometrie" title="Algebraische Geometrie – German" lang="de" hreflang="de" data-title="Algebraische Geometrie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B9%CE%BA%CE%AE_%CE%B3%CE%B5%CF%89%CE%BC%CE%B5%CF%84%CF%81%CE%AF%CE%B1" title="Αλγεβρική γεωμετρία – Greek" lang="el" hreflang="el" data-title="Αλγεβρική γεωμετρία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Geometr%C3%ADa_algebraica" title="Geometría algebraica – Spanish" lang="es" hreflang="es" data-title="Geometría algebraica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Algebra_geometrio" title="Algebra geometrio – Esperanto" lang="eo" hreflang="eo" data-title="Algebra geometrio" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Geometria_aljebraiko" title="Geometria aljebraiko – Basque" lang="eu" hreflang="eu" data-title="Geometria aljebraiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%86%D8%AF%D8%B3%D9%87_%D8%AC%D8%A8%D8%B1%DB%8C" title="هندسه جبری – Persian" lang="fa" hreflang="fa" data-title="هندسه جبری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie_alg%C3%A9brique" title="Géométrie algébrique – French" lang="fr" hreflang="fr" data-title="Géométrie algébrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Geoim%C3%A9adracht_ailg%C3%A9abrach" title="Geoiméadracht ailgéabrach – Irish" lang="ga" hreflang="ga" data-title="Geoiméadracht ailgéabrach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Xeometr%C3%ADa_alx%C3%A9brica" title="Xeometría alxébrica – Galician" lang="gl" hreflang="gl" data-title="Xeometría alxébrica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EC%88%98%EA%B8%B0%ED%95%98%ED%95%99" title="대수기하학 – Korean" lang="ko" hreflang="ko" data-title="대수기하학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%80%D5%A1%D5%B6%D6%80%D5%A1%D5%B0%D5%A1%D5%B7%D5%BE%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A5%D6%80%D5%AF%D6%80%D5%A1%D5%B9%D5%A1%D6%83%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Հանրահաշվական երկրաչափություն – Armenian" lang="hy" hreflang="hy" data-title="Հանրահաշվական երկրաչափություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A5%80%E0%A4%9C%E0%A5%80%E0%A4%AF_%E0%A4%9C%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%A4%E0%A4%BF" title="बीजीय ज्यामिति – Hindi" lang="hi" hreflang="hi" data-title="बीजीय ज्यामिति" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Algebarska_geometrija" title="Algebarska geometrija – Croatian" lang="hr" hreflang="hr" data-title="Algebarska geometrija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Geometri_aljabar" title="Geometri aljabar – Indonesian" lang="id" hreflang="id" data-title="Geometri aljabar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Geometria_algebrica" title="Geometria algebrica – Italian" lang="it" hreflang="it" data-title="Geometria algebrica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%92%D7%90%D7%95%D7%9E%D7%98%D7%A8%D7%99%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="גאומטריה אלגברית – Hebrew" lang="he" hreflang="he" data-title="גאומטריה אלגברית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%90%E1%83%9A%E1%83%92%E1%83%94%E1%83%91%E1%83%A0%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%92%E1%83%94%E1%83%9D%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%90" title="ალგებრული გეომეტრია – Georgian" lang="ka" hreflang="ka" data-title="ალგებრული გეომეტრია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%BB%D1%8B%D2%9B_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебралық геометрия – Kazakh" lang="kk" hreflang="kk" data-title="Алгебралық геометрия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%BB%D1%8B%D0%BA_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебралык геометрия – Kyrgyz" lang="ky" hreflang="ky" data-title="Алгебралык геометрия" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Algebrin%C4%97_geometrija" title="Algebrinė geometrija – Lithuanian" lang="lt" hreflang="lt" data-title="Algebrinė geometrija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Geometri_algebra" title="Geometri algebra – Malay" lang="ms" hreflang="ms" data-title="Geometri algebra" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%BB%D0%B8%D0%B3_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80" title="Алгебрлиг геометр – Mongolian" lang="mn" hreflang="mn" data-title="Алгебрлиг геометр" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%80%E1%80%B9%E1%80%81%E1%80%9B%E1%80%AC%E1%80%9E%E1%80%84%E1%80%BA%E1%80%B9%E1%80%81%E1%80%BB%E1%80%AC%E1%80%94%E1%80%8A%E1%80%BA%E1%80%B8%E1%80%80%E1%80%BB_%E1%80%82%E1%80%BB%E1%80%AE%E1%80%A9%E1%80%99%E1%80%B1%E1%80%90%E1%80%BC%E1%80%AE" title="အက္ခရာသင်္ချာနည်းကျ ဂျီဩမေတြီ – Burmese" lang="my" hreflang="my" data-title="အက္ခရာသင်္ချာနည်းကျ ဂျီဩမေတြီ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra%C3%AFsche_meetkunde" title="Algebraïsche meetkunde – Dutch" lang="nl" hreflang="nl" data-title="Algebraïsche meetkunde" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E5%B9%BE%E4%BD%95%E5%AD%A6" title="代数幾何学 – Japanese" lang="ja" hreflang="ja" data-title="代数幾何学" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Algebraisk_geometri" title="Algebraisk geometri – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Algebraisk geometri" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Algebraisk_geometri" title="Algebraisk geometri – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Algebraisk geometri" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Algebraik_geometriya" title="Algebraik geometriya – Uzbek" lang="uz" hreflang="uz" data-title="Algebraik geometriya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Geometria_algebraiczna" title="Geometria algebraiczna – Polish" lang="pl" hreflang="pl" data-title="Geometria algebraiczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Geometria_alg%C3%A9brica" title="Geometria algébrica – Portuguese" lang="pt" hreflang="pt" data-title="Geometria algébrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Geometrie_algebric%C4%83" title="Geometrie algebrică – Romanian" lang="ro" hreflang="ro" data-title="Geometrie algebrică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%8F" title="Алгебраическая геометрия – Russian" lang="ru" hreflang="ru" data-title="Алгебраическая геометрия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Giomitr%C3%ACa_alg%C3%A8bbrica" title="Giomitrìa algèbbrica – Sicilian" lang="scn" hreflang="scn" data-title="Giomitrìa algèbbrica" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Algebraic_geometry" title="Algebraic geometry – Simple English" lang="en-simple" hreflang="en-simple" data-title="Algebraic geometry" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Algebrick%C3%A1_geometria" title="Algebrická geometria – Slovak" lang="sk" hreflang="sk" data-title="Algebrická geometria" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Algebrska_geometrija" title="Algebrska geometrija – Slovenian" lang="sl" hreflang="sl" data-title="Algebrska geometrija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A6%DB%95%D9%86%D8%AF%D8%A7%D8%B2%DB%95%DB%8C_%D8%AC%DB%95%D8%A8%D8%B1%DB%8C" title="ئەندازەی جەبری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ئەندازەی جەبری" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Algebarska_geometrija" title="Algebarska geometrija – Serbian" lang="sr" hreflang="sr" data-title="Algebarska geometrija" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Algebrallinen_geometria" title="Algebrallinen geometria – Finnish" lang="fi" hreflang="fi" data-title="Algebrallinen geometria" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Algebraisk_geometri" title="Algebraisk geometri – Swedish" lang="sv" hreflang="sv" data-title="Algebraisk geometri" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Heometriyang_alhebraiko" title="Heometriyang alhebraiko – Tagalog" lang="tl" hreflang="tl" data-title="Heometriyang alhebraiko" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%AF%E0%AE%B1%E0%AF%8D%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4_%E0%AE%B5%E0%AE%9F%E0%AE%BF%E0%AE%B5%E0%AE%B5%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="இயற்கணித வடிவவியல் – Tamil" lang="ta" hreflang="ta" data-title="இயற்கணித வடிவவியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A3%E0%B8%82%E0%B8%B2%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%9E%E0%B8%B5%E0%B8%8A%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95" title="เรขาคณิตเชิงพีชคณิต – Thai" lang="th" hreflang="th" data-title="เรขาคณิตเชิงพีชคณิต" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D2%B2%D0%B0%D0%BD%D0%B4%D0%B0%D1%81%D0%B0%D0%B8_%D2%B7%D0%B0%D0%B1%D1%80%D3%A3" title="Ҳандасаи ҷабрӣ – Tajik" lang="tg" hreflang="tg" data-title="Ҳандасаи ҷабрӣ" data-language-autonym="Тоҷикӣ" data-language-local-name="Tajik" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Cebirsel_geometri" title="Cebirsel geometri – Turkish" lang="tr" hreflang="tr" data-title="Cebirsel geometri" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%B3%D0%B5%D0%BE%D0%BC%D0%B5%D1%82%D1%80%D1%96%D1%8F" title="Алгебрична геометрія – Ukrainian" lang="uk" hreflang="uk" data-title="Алгебрична геометрія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/H%C3%ACnh_h%E1%BB%8Dc_%C4%91%E1%BA%A1i_s%E1%BB%91" title="Hình học đại số – Vietnamese" lang="vi" hreflang="vi" data-title="Hình học đại số" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E5%87%A0%E4%BD%95" title="代数几何 – Wu" lang="wuu" hreflang="wuu" data-title="代数几何" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E4%BB%A3%E6%95%B8%E5%B9%BE%E4%BD%95" title="代數幾何 – Cantonese" lang="yue" hreflang="yue" data-title="代數幾何" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E5%87%A0%E4%BD%95" title="代数几何 – Chinese" lang="zh" hreflang="zh" data-title="代数几何" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Branch of mathematics</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about a branch of mathematics. For the book by Robin Hartshorne, see <a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry (book)</a>. For the journal, see <a href="/wiki/Algebraic_Geometry_(journal)" class="mw-redirect" title="Algebraic Geometry (journal)">Algebraic Geometry (journal)</a>. 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Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Algebraic+geometry%22">"Algebraic geometry"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Algebraic+geometry%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Algebraic+geometry%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Algebraic+geometry%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Algebraic+geometry%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Algebraic+geometry%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">January 2020</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Togliatti_surface.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Togliatti_surface.png/220px-Togliatti_surface.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Togliatti_surface.png/330px-Togliatti_surface.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Togliatti_surface.png/440px-Togliatti_surface.png 2x" data-file-width="800" data-file-height="800" /></a><figcaption>This <a href="/wiki/Togliatti_surface" title="Togliatti surface">Togliatti surface</a> is an <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surface</a> of degree five. 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class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th></tr><tr><td class="sidebar-image"><span class="mw-default-size notpageimage" typeof="mw:File/Frameless"><a href="/wiki/File:Stereographic_projection_in_3D.svg" class="mw-file-description"><img alt="Stereographic projection from the top of a sphere onto a plane beneath it" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/220px-Stereographic_projection_in_3D.svg.png" decoding="async" width="220" height="162" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/330px-Stereographic_projection_in_3D.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/88/Stereographic_projection_in_3D.svg/440px-Stereographic_projection_in_3D.svg.png 2x" data-file-width="870" data-file-height="639" /></a></span><div class="sidebar-caption"><a href="/wiki/Projective_geometry" title="Projective geometry">Projecting</a> a <a href="/wiki/Sphere" title="Sphere">sphere</a> to a <a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">Branches</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean</a> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a> <ul><li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li></ul></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li></ul></li> <li><a href="/wiki/Non-Archimedean_geometry" title="Non-Archimedean geometry">Non-Archimedean geometry</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Synthetic_geometry" title="Synthetic geometry">Synthetic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a class="mw-selflink selflink">Algebraic</a> <ul><li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine</a></li></ul></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a> <ul><li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Discrete_differential_geometry" title="Discrete differential geometry">Discrete differential</a></li></ul></li> <li><a href="/wiki/Complex_geometry" title="Complex geometry">Complex</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete/Combinatorial</a> <ul><li><a href="/wiki/Digital_geometry" title="Digital geometry">Digital</a></li></ul></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Computational_geometry" title="Computational geometry">Computational</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Incidence_geometry" title="Incidence geometry">Incidence </a></li> <li><a href="/wiki/Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a> <ul><li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>Concepts</li><li>Features</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content hlist"><a href="/wiki/Dimension_(geometry)" class="mw-redirect" title="Dimension (geometry)">Dimension</a> <ul><li><a href="/wiki/Straightedge_and_compass_construction" title="Straightedge and compass construction">Straightedge and compass constructions</a></li></ul> <ul><li><a href="/wiki/Angle" title="Angle">Angle</a></li> <li><a href="/wiki/Curve" title="Curve">Curve</a></li> <li><a href="/wiki/Diagonal" title="Diagonal">Diagonal</a></li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Perpendicular" title="Perpendicular">Perpendicular</a>)</li> <li><a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">Parallel</a></li> <li><a href="/wiki/Vertex_(geometry)" title="Vertex (geometry)">Vertex</a></li></ul> <ul><li><a href="/wiki/Congruence_(geometry)" title="Congruence (geometry)">Congruence</a></li> <li><a href="/wiki/Similarity_(geometry)" title="Similarity (geometry)">Similarity</a></li> <li><a href="/wiki/Symmetry" title="Symmetry">Symmetry</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Zero-dimensional_space" title="Zero-dimensional space">Zero-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Point_(geometry)" title="Point (geometry)">Point</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/One-dimensional_space" title="One-dimensional space">One-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Line_(geometry)" title="Line (geometry)">Line</a> <ul><li><a href="/wiki/Line_segment" title="Line segment">segment</a></li> <li><a href="/wiki/Line_(geometry)#Ray" title="Line (geometry)">ray</a></li></ul></li> <li><a href="/wiki/Length" title="Length">Length</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Two-dimensional_space" title="Two-dimensional space">Two-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane</a></li> <li><a href="/wiki/Area" title="Area">Area</a></li> <li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Triangle" title="Triangle">Triangle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Altitude_(triangle)" title="Altitude (triangle)">Altitude</a></li> <li><a href="/wiki/Hypotenuse" title="Hypotenuse">Hypotenuse</a></li> <li><a href="/wiki/Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Parallelogram" title="Parallelogram">Parallelogram</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Square" title="Square">Square</a></li> <li><a href="/wiki/Rectangle" title="Rectangle">Rectangle</a></li> <li><a href="/wiki/Rhombus" title="Rhombus">Rhombus</a></li> <li><a href="/wiki/Rhomboid" title="Rhomboid">Rhomboid</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Quadrilateral" title="Quadrilateral">Quadrilateral</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Trapezoid" title="Trapezoid">Trapezoid</a></li> <li><a href="/wiki/Kite_(geometry)" title="Kite (geometry)">Kite</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Circle" title="Circle">Circle</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Diameter" title="Diameter">Diameter</a></li> <li><a href="/wiki/Circumference" title="Circumference">Circumference</a></li> <li><a href="/wiki/Area_of_a_circle" title="Area of a circle">Area</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Three-dimensional_space" title="Three-dimensional space">Three-dimensional</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Volume" title="Volume">Volume</a></li></ul> <ul><li><a href="/wiki/Cube" title="Cube">Cube</a> <ul><li><a href="/wiki/Cuboid" title="Cuboid">cuboid</a></li></ul></li> <li><a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">Cylinder</a></li> <li><a href="/wiki/Dodecahedron" title="Dodecahedron">Dodecahedron</a></li> <li><a href="/wiki/Icosahedron" title="Icosahedron">Icosahedron</a></li> <li><a href="/wiki/Octahedron" title="Octahedron">Octahedron</a></li> <li><a href="/wiki/Pyramid_(geometry)" title="Pyramid (geometry)">Pyramid</a></li> <li><a href="/wiki/Platonic_Solid" class="mw-redirect" title="Platonic Solid">Platonic Solid</a></li> <li><a href="/wiki/Sphere" title="Sphere">Sphere</a></li> <li><a href="/wiki/Tetrahedron" title="Tetrahedron">Tetrahedron</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Four-dimensional_space" title="Four-dimensional space">Four</a>- / other-dimensional</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Tesseract" title="Tesseract">Tesseract</a></li> <li><a href="/wiki/Hypersphere" class="mw-redirect" title="Hypersphere">Hypersphere</a></li></ul></div></div></td> </tr><tr><th class="sidebar-heading" style="padding-bottom:0.2em;"> <a href="/wiki/List_of_geometers" title="List of geometers">Geometers</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by name</div></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/List_of_geometers" title="List of geometers">List of geometers</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content-with-subgroup"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf; text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">by period</div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="padding-bottom:0;"><table class="sidebar-subgroup"><tbody><tr><th class="sidebar-heading"> <a href="/wiki/Before_Common_Era" class="mw-redirect" title="Before Common Era">BCE</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Ahmes" title="Ahmes">Ahmes</a></li> <li><a href="/wiki/Baudhayana" class="mw-redirect" title="Baudhayana">Baudhayana</a></li> <li><a href="/wiki/Manava" title="Manava">Manava</a></li> <li><a href="/wiki/Pythagoras" title="Pythagoras">Pythagoras</a></li> <li><a href="/wiki/Euclid" title="Euclid">Euclid</a></li> <li><a href="/wiki/Archimedes" title="Archimedes">Archimedes</a></li> <li><a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1–1400s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Zhang_Heng" title="Zhang Heng">Zhang</a></li> <li><a href="/wiki/K%C4%81ty%C4%81yana" title="Kātyāyana">Kātyāyana</a></li> <li><a href="/wiki/Aryabhata" title="Aryabhata">Aryabhata</a></li> <li><a href="/wiki/Brahmagupta" title="Brahmagupta">Brahmagupta</a></li> <li><a href="/wiki/Virasena" title="Virasena">Virasena</a></li> <li><a href="/wiki/Alhazen" class="mw-redirect" title="Alhazen">Alhazen</a></li> <li><a href="/wiki/Sijzi" class="mw-redirect" title="Sijzi">Sijzi</a></li> <li><a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Khayyám</a></li> <li><a href="/wiki/Ibn_al-Yasamin" title="Ibn al-Yasamin">al-Yasamin</a></li> <li><a href="/wiki/Nasir_al-Din_al-Tusi" title="Nasir al-Din al-Tusi">al-Tusi</a></li> <li><a href="/wiki/Yang_Hui" title="Yang Hui">Yang Hui</a></li> <li><a href="/wiki/Parameshvara" class="mw-redirect" title="Parameshvara">Parameshvara</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1400s–1700s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Jye%E1%B9%A3%E1%B9%ADhadeva" title="Jyeṣṭhadeva">Jyeṣṭhadeva</a></li> <li><a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a></li> <li><a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Pascal</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Minggatu" title="Minggatu">Minggatu</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Sakabe_K%C5%8Dhan" title="Sakabe Kōhan">Sakabe</a></li> <li><a href="/wiki/Yasuaki_Aida" class="mw-redirect" title="Yasuaki Aida">Aida</a></li></ul></td> </tr><tr><th class="sidebar-heading"> 1700s–1900s</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Nikolai_Lobachevsky" title="Nikolai Lobachevsky">Lobachevsky</a></li> <li><a href="/wiki/J%C3%A1nos_Bolyai" title="János Bolyai">Bolyai</a></li> <li><a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Riemann</a></li> <li><a href="/wiki/Felix_Klein" title="Felix Klein">Klein</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a></li> <li><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a></li> <li><a href="/wiki/Hermann_Minkowski" title="Hermann Minkowski">Minkowski</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Cartan</a></li> <li><a href="/wiki/Oswald_Veblen" title="Oswald Veblen">Veblen</a></li> <li><a href="/wiki/Harold_Scott_MacDonald_Coxeter" title="Harold Scott MacDonald Coxeter">Coxeter</a></li> <li><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Chern</a></li></ul></td> </tr><tr><th class="sidebar-heading"> Present day</th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Mikhail_Leonidovich_Gromov" class="mw-redirect" title="Mikhail Leonidovich Gromov">Gromov</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:General_geometry" title="Template:General geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:General_geometry" title="Template talk:General geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:General_geometry" title="Special:EditPage/Template:General geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Algebraic geometry</b> is a branch of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a> which uses <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebraic</a> techniques, mainly from <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, to solve <a href="/wiki/Geometry" title="Geometry">geometrical problems</a>. Classically, it studies <a href="/wiki/Zero_of_a_function" title="Zero of a function">zeros</a> of <a href="/wiki/Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">multivariate polynomials</a>; the modern approach generalizes this in a few different aspects. </p><p>The fundamental objects of study in algebraic geometry are <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a>, which are geometric manifestations of <a href="/wiki/Solution_set" title="Solution set">solutions</a> of <a href="/wiki/Systems_of_polynomial_equations" class="mw-redirect" title="Systems of polynomial equations">systems of polynomial equations</a>. Examples of the most studied classes of algebraic varieties are <a href="/wiki/Line_(geometry)" title="Line (geometry)">lines</a>, <a href="/wiki/Circle" title="Circle">circles</a>, <a href="/wiki/Parabola" title="Parabola">parabolas</a>, <a href="/wiki/Ellipse" title="Ellipse">ellipses</a>, <a href="/wiki/Hyperbola" title="Hyperbola">hyperbolas</a>, <a href="/wiki/Cubic_curve" class="mw-redirect" title="Cubic curve">cubic curves</a> like <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a>, and quartic curves like <a href="/wiki/Lemniscate_of_Bernoulli" title="Lemniscate of Bernoulli">lemniscates</a> and <a href="/wiki/Cassini_oval" title="Cassini oval">Cassini ovals</a>. These are <a href="/wiki/Plane_algebraic_curve" class="mw-redirect" title="Plane algebraic curve">plane algebraic curves</a>. A point of the plane lies on an algebraic curve if its coordinates satisfy a given <a href="/wiki/Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equation</a>. Basic questions involve the study of points of special interest like <a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">singular points</a>, <a href="/wiki/Inflection_point" title="Inflection point">inflection points</a> and <a href="/wiki/Point_at_infinity" title="Point at infinity">points at infinity</a>. More advanced questions involve the <a href="/wiki/Topology" title="Topology">topology</a> of the curve and the relationship between curves defined by different equations. </p><p>Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, topology and <a href="/wiki/Number_theory" title="Number theory">number theory</a>. As a study of systems of polynomial equations in several variables, the subject of algebraic geometry begins with finding specific solutions via <a href="/wiki/Equation_solving" title="Equation solving">equation solving</a>, and then proceeds to understand the intrinsic properties of the totality of solutions of a system of equations. This understanding requires both conceptual theory and computational technique. </p><p>In the 20th century, algebraic geometry split into several subareas. </p> <ul><li>The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a>.</li> <li><a href="/wiki/Real_algebraic_geometry" title="Real algebraic geometry">Real algebraic geometry</a> is the study of the real algebraic varieties.</li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a> and, more generally, <a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">arithmetic geometry</a> is the study of algebraic varieties over <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> that are not algebraically closed and, specifically, over fields of interest in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, such as the field of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number fields</a>, <a href="/wiki/Finite_field" title="Finite field">finite fields</a>, <a href="/wiki/Algebraic_function_field" title="Algebraic function field">function fields</a>, and <a href="/wiki/P-adic_number" title="P-adic number"><i>p</i>-adic fields</a>.</li> <li>A large part of <a href="/wiki/Singularity_theory#Singularities_in_algebraic_geometry" title="Singularity theory">singularity theory</a> is devoted to the singularities of algebraic varieties.</li> <li><a href="#Computational_algebraic_geometry">Computational algebraic geometry</a> is an area that has emerged at the intersection of algebraic geometry and <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a>, with the rise of computers. It consists mainly of <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> design and <a href="/wiki/Software" title="Software">software</a> development for the study of properties of explicitly given algebraic varieties.</li></ul> <p>Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space; this parallels developments in topology, <a href="/wiki/Differential_geometry" title="Differential geometry">differential</a> and <a href="/wiki/Complex_geometry" title="Complex geometry">complex geometry</a>. One key achievement of this abstract algebraic geometry is <a href="/wiki/Grothendieck" class="mw-redirect" title="Grothendieck">Grothendieck</a>'s <a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a> which allows one to use <a href="/wiki/Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a> to study algebraic varieties in a way which is very similar to its use in the study of <a href="/wiki/Differential_manifold" class="mw-redirect" title="Differential manifold">differential</a> and <a href="/wiki/Complex_manifold" title="Complex manifold">analytic manifolds</a>. This is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>, with a <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> of the <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a>, while the points of the corresponding affine scheme are all prime ideals of this ring. This means that a point of such a scheme may be either a usual point or a subvariety. This approach also enables a unification of the language and the tools of classical algebraic geometry, mainly concerned with complex points, and of algebraic number theory. <a href="/wiki/Wiles%27s_proof_of_Fermat%27s_Last_Theorem" title="Wiles's proof of Fermat's Last Theorem">Wiles' proof</a> of the longstanding conjecture called <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> is an example of the power of this approach. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_notions">Basic notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=1" title="Edit section: Basic notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Algebraic_variety" title="Algebraic variety">Algebraic variety</a></div> <div class="mw-heading mw-heading3"><h3 id="Zeros_of_simultaneous_polynomials">Zeros of simultaneous polynomials</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=2" title="Edit section: Zeros of simultaneous polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Slanted_circle.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/90/Slanted_circle.png/220px-Slanted_circle.png" decoding="async" width="220" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/9/90/Slanted_circle.png 1.5x" data-file-width="322" data-file-height="330" /></a><figcaption>Sphere and slanted circle</figcaption></figure> <p>In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, meaning the set of all points that simultaneously satisfy one or more <a href="/wiki/Systems_of_polynomial_equations" class="mw-redirect" title="Systems of polynomial equations">polynomial equations</a>. For instance, the <a href="/wiki/N-sphere" title="N-sphere">two-dimensional</a> <a href="/wiki/Sphere" title="Sphere">sphere</a> of radius 1 in three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup>3</sup> could be defined as the set of all points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> with </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}-1=0.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}-1=0.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e104923a7ad8516f60a5b0257a3e2ad7528378ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.722ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}-1=0.\,}"></span></dd></dl> <p>A "slanted" circle in <b>R</b><sup>3</sup> can be defined as the set of all points <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x,y,z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22a8c93372e8f8b6e24d523bd5545aed3430baf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.45ex; height:2.843ex;" alt="{\displaystyle (x,y,z)}"></span> which satisfy the two polynomial equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}+z^{2}-1=0,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}+z^{2}-1=0,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7db6d9ee74ec6ce4bb223f1a210d5e19c5ad3942" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.722ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}+z^{2}-1=0,\,}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y+z=0.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>0.</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y+z=0.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c335caf234a007ac8f51d22ef263cbac54fbda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.549ex; height:2.509ex;" alt="{\displaystyle x+y+z=0.\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Affine_varieties">Affine varieties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=3" title="Edit section: Affine varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Affine_variety" title="Affine variety">Affine variety</a></div> <p>First we start with a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>k</i>. In classical algebraic geometry, this field was always the complex numbers <b>C</b>, but many of the same results are true if we assume only that <i>k</i> is <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a>. We consider the <a href="/wiki/Affine_space" title="Affine space">affine space</a> of dimension <i>n</i> over <i>k</i>, denoted <b>A</b><sup>n</sup>(<i>k</i>) (or more simply <b>A</b><sup><i>n</i></sup>, when <i>k</i> is clear from the context). When one fixes a coordinate system, one may identify <b>A</b><sup>n</sup>(<i>k</i>) with <i>k</i><sup><i>n</i></sup>. The purpose of not working with <i>k</i><sup><i>n</i></sup> is to emphasize that one "forgets" the vector space structure that <i>k</i><sup>n</sup> carries. </p><p>A function <i>f</i> : <b>A</b><sup><i>n</i></sup> → <b>A</b><sup>1</sup> is said to be <i>polynomial</i> (or <i>regular</i>) if it can be written as a polynomial, that is, if there is a polynomial <i>p</i> in <i>k</i>[<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>] such that <i>f</i>(<i>M</i>) = <i>p</i>(<i>t</i><sub>1</sub>,...,<i>t</i><sub><i>n</i></sub>) for every point <i>M</i> with coordinates (<i>t</i><sub>1</sub>,...,<i>t</i><sub><i>n</i></sub>) in <b>A</b><sup><i>n</i></sup>. The property of a function to be polynomial (or regular) does not depend on the choice of a coordinate system in <b>A</b><sup><i>n</i></sup>. </p><p>When a coordinate system is chosen, the regular functions on the affine <i>n</i>-space may be identified with the ring of <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> in <i>n</i> variables over <i>k</i>. Therefore, the set of the regular functions on <b>A</b><sup><i>n</i></sup> is a ring, which is denoted <i>k</i>[<b>A</b><sup><i>n</i></sup>]. </p><p>We say that a polynomial <i>vanishes</i> at a point if evaluating it at that point gives zero. Let <i>S</i> be a set of polynomials in <i>k</i>[<b>A</b><sup>n</sup>]. The <i>vanishing set of S</i> (or <i>vanishing locus</i> or <i>zero set</i>) is the set <i>V</i>(<i>S</i>) of all points in <b>A</b><sup><i>n</i></sup> where every polynomial in <i>S</i> vanishes. Symbolically, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(S)=\{(t_{1},\dots ,t_{n})\mid p(t_{1},\dots ,t_{n})=0{\text{ for all }}p\in S\}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>p</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>p</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(S)=\{(t_{1},\dots ,t_{n})\mid p(t_{1},\dots ,t_{n})=0{\text{ for all }}p\in S\}.\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/515746f0489557e9f4bfa56da998d9ad398235c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.293ex; height:2.843ex;" alt="{\displaystyle V(S)=\{(t_{1},\dots ,t_{n})\mid p(t_{1},\dots ,t_{n})=0{\text{ for all }}p\in S\}.\,}"></span></dd></dl> <p>A subset of <b>A</b><sup><i>n</i></sup> which is <i>V</i>(<i>S</i>), for some <i>S</i>, is called an <i>algebraic set</i>. The <i>V</i> stands for <i>variety</i> (a specific type of algebraic set to be defined below). </p><p>Given a subset <i>U</i> of <b>A</b><sup><i>n</i></sup>, can one recover the set of polynomials which generate it? If <i>U</i> is <i>any</i> subset of <b>A</b><sup><i>n</i></sup>, define <i>I</i>(<i>U</i>) to be the set of all polynomials whose vanishing set contains <i>U</i>. The <i>I</i> stands for <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a>: if two polynomials <i>f</i> and <i>g</i> both vanish on <i>U</i>, then <i>f</i>+<i>g</i> vanishes on <i>U</i>, and if <i>h</i> is any polynomial, then <i>hf</i> vanishes on <i>U</i>, so <i>I</i>(<i>U</i>) is always an ideal of the polynomial ring <i>k</i>[<b>A</b><sup><i>n</i></sup>]. </p><p>Two natural questions to ask are: </p> <ul><li>Given a subset <i>U</i> of <b>A</b><sup><i>n</i></sup>, when is <i>U</i> = <i>V</i>(<i>I</i>(<i>U</i>))?</li> <li>Given a set <i>S</i> of polynomials, when is <i>S</i> = <i>I</i>(<i>V</i>(<i>S</i>))?</li></ul> <p>The answer to the first question is provided by introducing the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a>, a topology on <b>A</b><sup><i>n</i></sup> whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of <i>k</i>[<b>A</b><sup><i>n</i></sup>]. Then <i>U</i> = <i>V</i>(<i>I</i>(<i>U</i>)) if and only if <i>U</i> is an algebraic set or equivalently a Zariski-closed set. The answer to the second question is given by <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>. In one of its forms, it says that <i>I</i>(<i>V</i>(<i>S</i>)) is the <a href="/wiki/Radical_of_an_ideal" title="Radical of an ideal">radical</a> of the ideal generated by <i>S</i>. In more abstract language, there is a <a href="/wiki/Galois_connection" title="Galois connection">Galois connection</a>, giving rise to two <a href="/wiki/Closure_operator" title="Closure operator">closure operators</a>; they can be identified, and naturally play a basic role in the theory; the <a href="/wiki/Galois_connection#Examples" title="Galois connection">example</a> is elaborated at Galois connection. </p><p>For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set <i>U</i>. <a href="/wiki/Hilbert%27s_basis_theorem" title="Hilbert's basis theorem">Hilbert's basis theorem</a> implies that ideals in <i>k</i>[<b>A</b><sup><i>n</i></sup>] are always finitely generated. </p><p>An algebraic set is called <i><a href="/wiki/Irreducible_component" title="Irreducible component">irreducible</a></i> if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a finite union of irreducible algebraic sets and this decomposition is unique. Thus its elements are called the <i>irreducible components</i> of the algebraic set. An irreducible algebraic set is also called a <i><a href="/wiki/Algebraic_variety" title="Algebraic variety">variety</a></i>. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> of the polynomial ring. </p><p>Some authors do not make a clear distinction between algebraic sets and varieties and use <i>irreducible variety</i> to make the distinction when needed. </p> <div class="mw-heading mw-heading3"><h3 id="Regular_functions">Regular functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=4" title="Edit section: Regular functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Regular_function" class="mw-redirect" title="Regular function">Regular function</a></div> <p>Just as <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a> are the natural maps on <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> and <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> are the natural maps on <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>, there is a natural class of functions on an algebraic set, called <i>regular functions</i> or <i>polynomial functions</i>. A regular function on an algebraic set <i>V</i> contained in <b>A</b><sup><i>n</i></sup> is the restriction to <i>V</i> of a regular function on <b>A</b><sup><i>n</i></sup>. For an algebraic set defined on the field of the complex numbers, the regular functions are smooth and even <a href="/wiki/Analytic_function" title="Analytic function">analytic</a>. </p><p>It may seem unnaturally restrictive to require that a regular function always extend to the ambient space, but it is very similar to the situation in a <a href="/wiki/Normal_space" title="Normal space">normal</a> <a href="/wiki/Topological_space" title="Topological space">topological space</a>, where the <a href="/wiki/Tietze_extension_theorem" title="Tietze extension theorem">Tietze extension theorem</a> guarantees that a continuous function on a closed subset always extends to the ambient topological space. </p><p>Just as with the regular functions on affine space, the regular functions on <i>V</i> form a ring, which we denote by <i>k</i>[<i>V</i>]. This ring is called the <i><a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> of V</i>. </p><p>Since regular functions on V come from regular functions on <b>A</b><sup><i>n</i></sup>, there is a relationship between the coordinate rings. Specifically, if a regular function on <i>V</i> is the restriction of two functions <i>f</i> and <i>g</i> in <i>k</i>[<b>A</b><sup><i>n</i></sup>], then <i>f</i> − <i>g</i> is a polynomial function which is null on <i>V</i> and thus belongs to <i>I</i>(<i>V</i>). Thus <i>k</i>[<i>V</i>] may be identified with <i>k</i>[<b>A</b><sup><i>n</i></sup>]/<i>I</i>(<i>V</i>). </p> <div class="mw-heading mw-heading3"><h3 id="Morphism_of_affine_varieties">Morphism of affine varieties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=5" title="Edit section: Morphism of affine varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using regular functions from an affine variety to <b>A</b><sup>1</sup>, we can define <a href="/wiki/Morphism_of_algebraic_varieties" title="Morphism of algebraic varieties">regular maps</a> from one affine variety to another. First we will define a regular map from a variety into affine space: Let <i>V</i> be a variety contained in <b>A</b><sup><i>n</i></sup>. Choose <i>m</i> regular functions on <i>V</i>, and call them <i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>m</i></sub>. We define a <i>regular map</i> <i>f</i> from <i>V</i> to <b>A</b><sup><i>m</i></sup> by letting <span class="nowrap"><i>f</i> = (<i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>m</i></sub>)</span>. In other words, each <i>f</i><sub><i>i</i></sub> determines one coordinate of the <a href="/wiki/Image_(mathematics)" title="Image (mathematics)">range</a> of <i>f</i>. </p><p>If <i>V</i>′ is a variety contained in <b>A</b><sup><i>m</i></sup>, we say that <i>f</i> is a <i>regular map</i> from <i>V</i> to <i>V</i>′ if the range of <i>f</i> is contained in <i>V</i>′. </p><p>The definition of the regular maps apply also to algebraic sets. The regular maps are also called <i>morphisms</i>, as they make the collection of all affine algebraic sets into a <a href="/wiki/Category_theory" title="Category theory">category</a>, where the objects are the affine algebraic sets and the <a href="/wiki/Morphism" title="Morphism">morphisms</a> are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. </p><p>Given a regular map <i>g</i> from <i>V</i> to <i>V</i>′ and a regular function <i>f</i> of <i>k</i>[<i>V</i>′], then <span class="nowrap"><i>f</i> ∘ <i>g</i> ∈ <i>k</i>[<i>V</i>]</span>. The map <span class="nowrap"><i>f</i> → <i>f</i> ∘ <i>g</i></span> is a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from <i>k</i>[<i>V</i>′] to <i>k</i>[<i>V</i>]. Conversely, every ring homomorphism from <i>k</i>[<i>V</i>′] to <i>k</i>[<i>V</i>] defines a regular map from <i>V</i> to <i>V</i>′. This defines an <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalence of categories</a> between the category of algebraic sets and the <a href="/wiki/Dual_(category_theory)" title="Dual (category theory)">opposite category</a> of the finitely generated <a href="/wiki/Reduced_ring" title="Reduced ring">reduced</a> <i>k</i>-algebras. This equivalence is one of the starting points of <a href="/wiki/Scheme_theory" class="mw-redirect" title="Scheme theory">scheme theory</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Rational_function_and_birational_equivalence">Rational function and birational equivalence</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=6" title="Edit section: Rational function and birational equivalence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Rational_mapping" title="Rational mapping">Rational mapping</a></div> <p>In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. </p><p>If <i>V</i> is an affine variety, its coordinate ring is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> and has thus a <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> which is denoted <i>k</i>(<i>V</i>) and called the <i>field of the rational functions</i> on <i>V</i> or, shortly, the <i><a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a></i> of <i>V</i>. Its elements are the restrictions to <i>V</i> of the <a href="/wiki/Rational_function" title="Rational function">rational functions</a> over the affine space containing <i>V</i>. The <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> of a rational function <i>f</i> is not <i>V</i> but the <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a> of the subvariety (a hypersurface) where the denominator of <i>f</i> vanishes. </p><p>As with regular maps, one may define a <i>rational map</i> from a variety <i>V</i> to a variety <i>V</i>'. As with the regular maps, the rational maps from <i>V</i> to <i>V</i>' may be identified to the <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">field homomorphisms</a> from <i>k</i>(<i>V</i>') to <i>k</i>(<i>V</i>). </p><p>Two affine varieties are <i>birationally equivalent</i> if there are two rational functions between them which are <a href="/wiki/Function_inverse" class="mw-redirect" title="Function inverse">inverse</a> one to the other in the regions where both are defined. Equivalently, they are birationally equivalent if their function fields are isomorphic. </p><p>An affine variety is a <i><a href="/wiki/Rational_variety" title="Rational variety">rational variety</a></i> if it is birationally equivalent to an affine space. This means that the variety admits a <i>rational parameterization</i>, that is a <a href="/wiki/Parametrization_(geometry)" title="Parametrization (geometry)">parametrization</a> with <a href="/wiki/Rational_function" title="Rational function">rational functions</a>. For example, the circle of equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee594b8851d760d0e2d44aba714907aca657b8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.703ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-1=0}"></span> is a rational curve, as it has the <a href="/wiki/Parametric_equation" title="Parametric equation">parametric equation</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\frac {2\,t}{1+t^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\frac {2\,t}{1+t^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d208b079aae8935893ac2f7a161ea682f00b151e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.161ex; height:5.676ex;" alt="{\displaystyle x={\frac {2\,t}{1+t^{2}}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y={\frac {1-t^{2}}{1+t^{2}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y={\frac {1-t^{2}}{1+t^{2}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51749329d32afc606b467370cf75a46942e3ff8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.021ex; height:6.176ex;" alt="{\displaystyle y={\frac {1-t^{2}}{1+t^{2}}}\,,}"></span></dd></dl> <p>which may also be viewed as a rational map from the line to the circle. </p><p>The problem of <a href="/wiki/Resolution_of_singularities" title="Resolution of singularities">resolution of singularities</a> is to know if every algebraic variety is birationally equivalent to a variety whose projective completion is nonsingular (see also <a href="/wiki/Smooth_completion" title="Smooth completion">smooth completion</a>). It was solved in the affirmative in characteristic 0 by <a href="/wiki/Heisuke_Hironaka" title="Heisuke Hironaka">Heisuke Hironaka</a> in 1964 and is yet unsolved in finite characteristic. </p> <div class="mw-heading mw-heading3"><h3 id="Projective_variety">Projective variety</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=7" title="Edit section: Projective variety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_geometry_of_projective_spaces" title="Algebraic geometry of projective spaces">Algebraic geometry of projective spaces</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Parabola_%26_cubic_curve_in_projective_space.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Parabola_%26_cubic_curve_in_projective_space.png/220px-Parabola_%26_cubic_curve_in_projective_space.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Parabola_%26_cubic_curve_in_projective_space.png/330px-Parabola_%26_cubic_curve_in_projective_space.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Parabola_%26_cubic_curve_in_projective_space.png/440px-Parabola_%26_cubic_curve_in_projective_space.png 2x" data-file-width="2000" data-file-height="1500" /></a><figcaption>Parabola (<span class="nowrap"><i>y</i> = <i>x</i><sup>2</sup></span>, red) and cubic (<span class="nowrap"><i>y</i> = <i>x</i><sup>3</sup></span>, blue) in projective space</figcaption></figure> <p>Just as the formulas for the roots of second, third, and fourth degree polynomials suggest extending real numbers to the more algebraically complete setting of the complex numbers, many properties of algebraic varieties suggest extending affine space to a more geometrically complete projective space. Whereas the complex numbers are obtained by adding the number <i>i</i>, a root of the polynomial <span class="nowrap"><i>x</i><sup>2</sup> + 1</span>, projective space is obtained by adding in appropriate points "at infinity", points where parallel lines may meet. </p><p>To see how this might come about, consider the variety <span class="nowrap"><i>V</i>(<i>y</i> − <i>x</i><sup>2</sup>)</span>. If we draw it, we get a <a href="/wiki/Parabola" title="Parabola">parabola</a>. As <i>x</i> goes to positive infinity, the slope of the line from the origin to the point (<i>x</i>, <i>x</i><sup>2</sup>) also goes to positive infinity. As <i>x</i> goes to negative infinity, the slope of the same line goes to negative infinity. </p><p>Compare this to the variety <i>V</i>(<i>y</i> − <i>x</i><sup>3</sup>). This is a <a href="/wiki/Cubic_curve" class="mw-redirect" title="Cubic curve">cubic curve</a>. As <i>x</i> goes to positive infinity, the slope of the line from the origin to the point (<i>x</i>, <i>x</i><sup>3</sup>) goes to positive infinity just as before. But unlike before, as <i>x</i> goes to negative infinity, the slope of the same line goes to positive infinity as well; the exact opposite of the parabola. So the behavior "at infinity" of <i>V</i>(<i>y</i> − <i>x</i><sup>3</sup>) is different from the behavior "at infinity" of <i>V</i>(<i>y</i> − <i>x</i><sup>2</sup>). </p><p>The consideration of the <i>projective completion</i> of the two curves, which is their prolongation "at infinity" in the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>, allows us to quantify this difference: the point at infinity of the parabola is a <a href="/wiki/Regular_point_of_an_algebraic_variety" class="mw-redirect" title="Regular point of an algebraic variety">regular point</a>, whose tangent is the <a href="/wiki/Line_at_infinity" title="Line at infinity">line at infinity</a>, while the point at infinity of the cubic curve is a <a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">cusp</a>. Also, both curves are rational, as they are parameterized by <i>x</i>, and the <a href="/wiki/Riemann-Roch_theorem_for_algebraic_curves" class="mw-redirect" title="Riemann-Roch theorem for algebraic curves">Riemann-Roch theorem</a> implies that the cubic curve must have a singularity, which must be at infinity, as all its points in the affine space are regular. </p><p>Thus many of the properties of algebraic varieties, including birational equivalence and all the topological properties, depend on the behavior "at infinity" and so it is natural to study the varieties in projective space. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, <a href="/wiki/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a> on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For these reasons, projective space plays a fundamental role in algebraic geometry. </p><p>Nowadays, the <i><a href="/wiki/Projective_space" title="Projective space">projective space</a></i> <b>P</b><sup><i>n</i></sup> of dimension <i>n</i> is usually defined as the set of the lines passing through a point, considered as the origin, in the affine space of dimension <span class="nowrap"><i>n</i> + 1</span>, or equivalently to the set of the vector lines in a vector space of dimension <span class="nowrap"><i>n</i> + 1</span>. When a coordinate system has been chosen in the space of dimension <span class="nowrap"><i>n</i> + 1</span>, all the points of a line have the same set of coordinates, up to the multiplication by an element of <i>k</i>. This defines the <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> of a point of <b>P</b><sup><i>n</i></sup> as a sequence of <span class="nowrap"><i>n</i> + 1</span> elements of the base field <i>k</i>, defined up to the multiplication by a nonzero element of <i>k</i> (the same for the whole sequence). </p><p>A polynomial in <span class="nowrap"><i>n</i> + 1</span> variables vanishes at all points of a line passing through the origin if and only if it is <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous</a>. In this case, one says that the polynomial <i>vanishes</i> at the corresponding point of <b>P</b><sup><i>n</i></sup>. This allows us to define a <i>projective algebraic set</i> in <b>P</b><sup><i>n</i></sup> as the set <span class="nowrap"><i>V</i>(<i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>k</i></sub>)</span>, where a finite set of homogeneous polynomials <span class="nowrap">{<i>f</i><sub>1</sub>, ..., <i>f</i><sub><i>k</i></sub>} </span> vanishes. Like for affine algebraic sets, there is a bijection between the projective algebraic sets and the reduced <a href="/wiki/Homogeneous_ideal" class="mw-redirect" title="Homogeneous ideal">homogeneous ideals</a> which define them. The <i>projective varieties</i> are the projective algebraic sets whose defining ideal is prime. In other words, a projective variety is a projective algebraic set, whose <a href="/wiki/Homogeneous_coordinate_ring" title="Homogeneous coordinate ring">homogeneous coordinate ring</a> is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>, the <i>projective coordinates ring</i> being defined as the quotient of the graded ring or the polynomials in <span class="nowrap"><i>n</i> + 1</span> variables by the homogeneous (reduced) ideal defining the variety. Every projective algebraic set may be uniquely decomposed into a finite union of projective varieties. </p><p>The only regular functions which may be defined properly on a projective variety are the constant functions. Thus this notion is not used in projective situations. On the other hand, the <i>field of the rational functions</i> or <i>function field </i> is a useful notion, which, similarly to the affine case, is defined as the set of the quotients of two homogeneous elements of the same degree in the homogeneous coordinate ring. </p> <div class="mw-heading mw-heading2"><h2 id="Real_algebraic_geometry">Real algebraic geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=8" title="Edit section: Real algebraic geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Real_algebraic_geometry" title="Real algebraic geometry">Real algebraic geometry</a></div> <p>Real algebraic geometry is the study of real algebraic varieties. </p><p>The fact that the field of the real numbers is an <a href="/wiki/Ordered_field" title="Ordered field">ordered field</a> cannot be ignored in such a study. For example, the curve of equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+y^{2}-a=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+y^{2}-a=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c613305193b3debdff2ea6268bdd170445d1bf13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.77ex; height:3.009ex;" alt="{\displaystyle x^{2}+y^{2}-a=0}"></span> is a circle if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f34a80ea013edb56e340b19550430a8b6dfd7b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a>0}"></span>, but has no real points if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d7ca60f6ed64b99649dcee21847295fedf206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a<0}"></span>. Real algebraic geometry also investigates, more broadly, <i>semi-algebraic sets</i>, which are the solutions of systems of polynomial inequalities. For example, neither branch of the <a href="/wiki/Hyperbola" title="Hyperbola">hyperbola</a> of equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa089ae542f18818ea8145eba8eeb97b8e337450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle xy-1=0}"></span> is a real algebraic variety. However, the branch in the first quadrant is a semi-algebraic set defined by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy-1=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy-1=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa089ae542f18818ea8145eba8eeb97b8e337450" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle xy-1=0}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/80d24be5f0eb4a9173da6038badc8659546021d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}"></span>. </p><p>One open problem in real algebraic geometry is the following part of <a href="/wiki/Hilbert%27s_sixteenth_problem" title="Hilbert's sixteenth problem">Hilbert's sixteenth problem</a>: Decide which respective positions are possible for the ovals of a nonsingular <a href="/wiki/Plane_curve" title="Plane curve">plane curve</a> of degree 8. </p> <div class="mw-heading mw-heading2"><h2 id="Computational_algebraic_geometry">Computational algebraic geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=9" title="Edit section: Computational algebraic geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held at <a href="/wiki/Marseille" title="Marseille">Marseille</a>, France, in June 1979. At this meeting, </p> <ul><li>Dennis S. Arnon showed that <a href="/wiki/George_E._Collins" title="George E. Collins">George E. Collins</a>'s <a href="/wiki/Cylindrical_algebraic_decomposition" title="Cylindrical algebraic decomposition">Cylindrical algebraic decomposition</a> (CAD) allows the computation of the topology of semi-algebraic sets,</li> <li><a href="/wiki/Bruno_Buchberger" title="Bruno Buchberger">Bruno Buchberger</a> presented <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner bases</a> and his algorithm to compute them,</li> <li><a href="/wiki/Daniel_Lazard" title="Daniel Lazard">Daniel Lazard</a> presented a new algorithm for solving systems of homogeneous polynomial equations with a <a href="/wiki/Computational_complexity" title="Computational complexity">computational complexity</a> which is essentially polynomial in the expected number of solutions and thus simply exponential in the number of the unknowns. This algorithm is strongly related with <a href="/wiki/Francis_Sowerby_Macaulay" title="Francis Sowerby Macaulay">Macaulay</a>'s <a href="/wiki/Resultant" title="Resultant">multivariate resultant</a>.</li></ul> <p>Since then, most results in this area are related to one or several of these items either by using or improving one of these algorithms, or by finding algorithms whose complexity is simply exponential in the number of the variables. </p><p>A body of mathematical theory complementary to symbolic methods called <a href="/wiki/Numerical_algebraic_geometry" title="Numerical algebraic geometry">numerical algebraic geometry</a> has been developed over the last several decades. The main computational method is <a href="/wiki/Homotopy_continuation" class="mw-redirect" title="Homotopy continuation">homotopy continuation</a>. This supports, for example, a model of <a href="/wiki/Floating_point" class="mw-redirect" title="Floating point">floating point</a> computation for solving problems of algebraic geometry. </p> <div class="mw-heading mw-heading3"><h3 id="Gröbner_basis"><span id="Gr.C3.B6bner_basis"></span>Gröbner basis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=10" title="Edit section: Gröbner basis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></div> <p>A <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a> is a system of <a href="/wiki/Ideal_(ring_theory)#Definitions_and_motivation" title="Ideal (ring theory)">generators</a> of a polynomial <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> whose computation allows the deduction of many properties of the affine algebraic variety defined by the ideal. </p><p>Given an ideal <i>I</i> defining an algebraic set <i>V</i>: </p> <ul><li><i>V</i> is empty (over an algebraically closed extension of the basis field), if and only if the Gröbner basis for any <a href="/wiki/Monomial_order" title="Monomial order">monomial ordering</a> is reduced to {1}.</li> <li>By means of the <a href="/wiki/Hilbert_series" class="mw-redirect" title="Hilbert series">Hilbert series</a> one may compute the <a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">dimension</a> and the <a href="/wiki/Degree_of_an_algebraic_variety" title="Degree of an algebraic variety">degree</a> of <i>V</i> from any Gröbner basis of <i>I</i> for a monomial ordering refining the total degree.</li> <li>If the dimension of <i>V</i> is 0, one may compute the points (finite in number) of <i>V</i> from any Gröbner basis of <i>I</i> (see <a href="/wiki/Systems_of_polynomial_equations" class="mw-redirect" title="Systems of polynomial equations">Systems of polynomial equations</a>).</li> <li>A Gröbner basis computation allows one to remove from <i>V</i> all irreducible components which are contained in a given hypersurface.</li> <li>A Gröbner basis computation allows one to compute the Zariski closure of the image of <i>V</i> by the projection on the <i>k</i> first coordinates, and the subset of the image where the projection is not proper.</li> <li>More generally Gröbner basis computations allow one to compute the Zariski closure of the image and the <a href="/wiki/Critical_point_(mathematics)" title="Critical point (mathematics)">critical points</a> of a rational function of <i>V</i> into another affine variety.</li></ul> <p>Gröbner basis computations do not allow one to compute directly the primary decomposition of <i>I</i> nor the prime ideals defining the irreducible components of <i>V</i>, but most algorithms for this involve Gröbner basis computation. The algorithms which are not based on Gröbner bases use <a href="/wiki/Regular_chain" title="Regular chain">regular chains</a> but may need Gröbner bases in some exceptional situations. </p><p>Gröbner bases are deemed to be difficult to compute. In fact they may contain, in the worst case, polynomials whose degree is doubly exponential in the number of variables and a number of polynomials which is also doubly exponential. However, this is only a worst case complexity, and the complexity bound of Lazard's algorithm of 1979 may frequently apply. <a href="/wiki/Faug%C3%A8re_F5_algorithm" class="mw-redirect" title="Faugère F5 algorithm">Faugère F5 algorithm</a> realizes this complexity, as it may be viewed as an improvement of Lazard's 1979 algorithm. It follows that the best implementations allow one to compute almost routinely with algebraic sets of degree more than 100. This means that, presently, the difficulty of computing a Gröbner basis is strongly related to the intrinsic difficulty of the problem. </p> <div class="mw-heading mw-heading3"><h3 id="Cylindrical_algebraic_decomposition_(CAD)"><span id="Cylindrical_algebraic_decomposition_.28CAD.29"></span>Cylindrical algebraic decomposition (CAD)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=11" title="Edit section: Cylindrical algebraic decomposition (CAD)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>CAD is an algorithm which was introduced in 1973 by G. Collins to implement with an acceptable complexity the <a href="/wiki/Tarski%E2%80%93Seidenberg_theorem" title="Tarski–Seidenberg theorem">Tarski–Seidenberg theorem</a> on <a href="/wiki/Quantifier_elimination" title="Quantifier elimination">quantifier elimination</a> over the real numbers. </p><p>This theorem concerns the formulas of the <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> whose <a href="/wiki/Atomic_formula" title="Atomic formula">atomic formulas</a> are polynomial equalities or inequalities between polynomials with real coefficients. These formulas are thus the formulas which may be constructed from the atomic formulas by the logical operators <i>and</i> (∧), <i>or</i> (∨), <i>not</i> (¬), <i>for all</i> (∀) and <i>exists</i> (∃). Tarski's theorem asserts that, from such a formula, one may compute an equivalent formula without quantifier (∀, ∃). </p><p>The complexity of CAD is doubly exponential in the number of variables. This means that CAD allows, in theory, to solve every problem of real algebraic geometry which may be expressed by such a formula, that is almost every problem concerning explicitly given varieties and semi-algebraic sets. </p><p>While Gröbner basis computation has doubly exponential complexity only in rare cases, CAD has almost always this high complexity. This implies that, unless if most polynomials appearing in the input are linear, it may not solve problems with more than four variables. </p><p>Since 1973, most of the research on this subject is devoted either to improving CAD or finding alternative algorithms in special cases of general interest. </p><p>As an example of the state of art, there are efficient algorithms to find at least a point in every connected component of a semi-algebraic set, and thus to test if a semi-algebraic set is empty. On the other hand, CAD is yet, in practice, the best algorithm to count the number of connected components. </p> <div class="mw-heading mw-heading3"><h3 id="Asymptotic_complexity_vs._practical_efficiency">Asymptotic complexity vs. practical efficiency</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=12" title="Edit section: Asymptotic complexity vs. practical efficiency"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The basic general algorithms of computational geometry have a double exponential worst case <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">complexity</a>. More precisely, if <i>d</i> is the maximal degree of the input polynomials and <i>n</i> the number of variables, their complexity is at most <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{2^{cn}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2^{cn}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b34a78e61f43372506ed92b5f50c87490602782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.815ex; height:2.676ex;" alt="{\displaystyle d^{2^{cn}}}"></span> for some constant <i>c</i>, and, for some inputs, the complexity is at least <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{2^{c'n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>c</mi> <mo>′</mo> </msup> <mi>n</mi> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{2^{c'n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1eb245f5b925f671c7bb44fd83676fdb86b4605" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.316ex; height:3.343ex;" alt="{\displaystyle d^{2^{c'n}}}"></span> for another constant <i>c</i>′. </p><p>During the last 20 years of the 20th century, various algorithms have been introduced to solve specific subproblems with a better complexity. Most of these algorithms have a complexity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{O(n^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{O(n^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1820f6f2c447eda6918fda45f00a3390fd3ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.801ex; height:3.009ex;" alt="{\displaystyle d^{O(n^{2})}}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Among these algorithms which solve a sub problem of the problems solved by Gröbner bases, one may cite <i>testing if an affine variety is empty</i> and <i>solving nonhomogeneous polynomial systems which have a finite number of solutions.</i> Such algorithms are rarely implemented because, on most entries <a href="/wiki/Faug%C3%A8re%27s_F4_and_F5_algorithms" title="Faugère's F4 and F5 algorithms">Faugère's F4 and F5 algorithms</a> have a better practical efficiency and probably a similar or better complexity (<i>probably</i> because the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been done only in a few special cases). </p><p>The main algorithms of real algebraic geometry which solve a problem solved by CAD are related to the topology of semi-algebraic sets. One may cite <i>counting the number of connected components</i>, <i>testing if two points are in the same components</i> or <i>computing a <a href="/wiki/Whitney_stratification" class="mw-redirect" title="Whitney stratification">Whitney stratification</a> of a real algebraic set</i>. They have a complexity of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d^{O(n^{2})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d^{O(n^{2})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1820f6f2c447eda6918fda45f00a3390fd3ef8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.801ex; height:3.009ex;" alt="{\displaystyle d^{O(n^{2})}}"></span>, but the constant involved by <i>O</i> notation is so high that using them to solve any nontrivial problem effectively solved by CAD, is impossible even if one could use all the existing computing power in the world. Therefore, these algorithms have never been implemented and this is an active research area to search for algorithms with have together a good asymptotic complexity and a good practical efficiency. </p> <div class="mw-heading mw-heading2"><h2 id="Abstract_modern_viewpoint">Abstract modern viewpoint</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=13" title="Edit section: Abstract modern viewpoint"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, <a href="/wiki/Formal_scheme" title="Formal scheme">formal schemes</a>, <a href="/wiki/Ind-scheme" title="Ind-scheme">ind-schemes</a>, <a href="/wiki/Algebraic_space" title="Algebraic space">algebraic spaces</a>, <a href="/wiki/Algebraic_stack" title="Algebraic stack">algebraic stacks</a> and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal functions of Zariski can be accommodated by introducing nilpotent elements in structure rings; considering spaces of loops and arcs, constructing quotients by group actions and developing formal grounds for natural <a href="/wiki/Intersection_theory" title="Intersection theory">intersection theory</a> and <a href="/wiki/Deformation_theory" class="mw-redirect" title="Deformation theory">deformation theory</a> lead to some of the further extensions. </p><p>Most remarkably, in the early 1960s, <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> were subsumed into <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a>'s concept of a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>. Their local objects are affine schemes or prime spectra which are locally ringed spaces which form a category which is antiequivalent to the category of commutative unital rings, extending the duality between the category of affine algebraic varieties over a field <i>k</i>, and the category of finitely generated reduced <i>k</i>-algebras. The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a>. Grothendieck introduced Grothendieck topologies having in mind more exotic but geometrically finer and more sensitive examples than the crude Zariski topology, namely the <a href="/wiki/%C3%89tale_topology" title="Étale topology">étale topology</a>, and the two flat Grothendieck topologies: fppf and fpqc; nowadays some other examples became prominent including <a href="/wiki/Nisnevich_topology" title="Nisnevich topology">Nisnevich topology</a>. Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to <a href="/wiki/Artin_stack" class="mw-redirect" title="Artin stack">Artin stacks</a> and, even finer, <a href="/wiki/Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stacks</a>, both often called algebraic stacks. </p><p>Sometimes other algebraic sites replace the category of affine schemes. For example, <a href="/wiki/Nikolai_Durov" title="Nikolai Durov">Nikolai Durov</a> has introduced commutative algebraic monads as a generalization of local objects in a generalized algebraic geometry. Versions of a <a href="/wiki/Tropical_geometry" title="Tropical geometry">tropical geometry</a>, of an <a href="/wiki/Absolute_geometry" title="Absolute geometry">absolute geometry</a> over a field of one element and an algebraic analogue of <a href="/wiki/Arakelov%27s_geometry" class="mw-redirect" title="Arakelov's geometry">Arakelov's geometry</a> were realized in this setup. </p><p>Another formal generalization is possible to <a href="/wiki/Universal_algebraic_geometry" title="Universal algebraic geometry">universal algebraic geometry</a> in which every <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">variety of algebras</a> has its own algebraic geometry. The term <i>variety of algebras</i> should not be confused with <i>algebraic variety</i>. </p><p>The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. </p><p>Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the most natural approach. One can extend the <a href="/wiki/Grothendieck_site" class="mw-redirect" title="Grothendieck site">Grothendieck site</a> of affine schemes to a <a href="/wiki/Higher_category_theory" title="Higher category theory">higher categorical</a> site of <a href="/wiki/Derived_affine_scheme" class="mw-redirect" title="Derived affine scheme">derived affine schemes</a>, by replacing the commutative rings with an infinity category of <a href="/wiki/Differential_graded_commutative_algebra" class="mw-redirect" title="Differential graded commutative algebra">differential graded commutative algebras</a>, or of simplicial commutative rings or a similar category with an appropriate variant of a Grothendieck topology. One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical version of a sheaf axiom (and to be algebraic, inductively a sequence of representability conditions). <a href="/wiki/Quillen_model_category" class="mw-redirect" title="Quillen model category">Quillen model categories</a>, Segal categories and <a href="/wiki/Quasicategory" class="mw-redirect" title="Quasicategory">quasicategories</a> are some of the most often used tools to formalize this yielding the <i><a href="/wiki/Derived_algebraic_geometry" title="Derived algebraic geometry">derived algebraic geometry</a></i>, introduced by the school of <a href="/wiki/Carlos_Simpson" title="Carlos Simpson">Carlos Simpson</a>, including Andre Hirschowitz, <a href="/wiki/Bertrand_To%C3%ABn" title="Bertrand Toën">Bertrand Toën</a>, Gabrielle Vezzosi, Michel Vaquié and others; and developed further by <a href="/wiki/Jacob_Lurie" title="Jacob Lurie">Jacob Lurie</a>, <a href="/wiki/Bertrand_To%C3%ABn" title="Bertrand Toën">Bertrand Toën</a>, and <a href="/wiki/Gabriele_Vezzosi" title="Gabriele Vezzosi">Gabriele Vezzosi</a>. Another (noncommutative) version of derived algebraic geometry, using A-infinity categories has been developed from the early 1990s by <a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a> and followers. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=14" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Before_the_16th_century">Before the 16th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=15" title="Edit section: Before the 16th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some of the roots of algebraic geometry date back to the work of the <a href="/wiki/Hellenistic_Greece" title="Hellenistic Greece">Hellenistic Greeks</a> from the 5th century BC. The <a href="/wiki/Delian_problem" class="mw-redirect" title="Delian problem">Delian problem</a>, for instance, was to construct a length <i>x</i> so that the cube of side <i>x</i> contained the same volume as the rectangular box <i>a</i><sup>2</sup><i>b</i> for given sides <i>a</i> and <i>b</i>. <a href="/wiki/Menaechmus" title="Menaechmus">Menaechmus</a> (<abbr title="circa">c.</abbr><span style="white-space:nowrap;"> 350 BC</span>) considered the problem geometrically by intersecting the pair of plane conics <i>ay</i> = <i>x</i><sup>2</sup> and <i>xy</i> = <i>ab</i>.<sup id="cite_ref-Dieudonné_2-0" class="reference"><a href="#cite_note-Dieudonné-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> In the 3rd century BC, <a href="/wiki/Archimedes" title="Archimedes">Archimedes</a> and <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a> systematically studied additional problems on <a href="/wiki/Conic_sections" class="mw-redirect" title="Conic sections">conic sections</a> using coordinates.<sup id="cite_ref-Dieudonné_2-1" class="reference"><a href="#cite_note-Dieudonné-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEKline1972108,_90_3-0" class="reference"><a href="#cite_note-FOOTNOTEKline1972108,_90-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Apollonius_of_Perga" title="Apollonius of Perga">Apollonius</a> in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">Descartes</a> by some 1800 years.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> His application of reference lines, a <a href="/wiki/Diameter" title="Diameter">diameter</a> and a <a href="/wiki/Tangent" title="Tangent">tangent</a> is essentially no different from our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding coordinates using geometric methods like using parabolas and curves.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Medieval mathematicians, including <a href="/wiki/Omar_Khayyam" title="Omar Khayyam">Omar Khayyam</a>, <a href="/wiki/Leonardo_of_Pisa" class="mw-redirect" title="Leonardo of Pisa">Leonardo of Pisa</a>, <a href="/wiki/Gersonides" title="Gersonides">Gersonides</a> and <a href="/wiki/Nicole_Oresme" title="Nicole Oresme">Nicole Oresme</a> in the <a href="/wiki/Middle_Ages" title="Middle Ages">Medieval Period</a> ,<sup id="cite_ref-FOOTNOTEKline1972193_8-0" class="reference"><a href="#cite_note-FOOTNOTEKline1972193-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> solved certain cubic and quadratic equations by purely algebraic means and then interpreted the results geometrically. The <a href="/wiki/Persian_people" class="mw-redirect" title="Persian people">Persian</a> mathematician <a href="/wiki/Omar_Khayy%C3%A1m" class="mw-redirect" title="Omar Khayyám">Omar Khayyám</a> (born 1048 AD) believed that there was a relationship between <a href="/wiki/Arithmetic" title="Arithmetic">arithmetic</a>, <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>.<sup id="cite_ref-FOOTNOTEKline1972193–195_9-0" class="reference"><a href="#cite_note-FOOTNOTEKline1972193–195-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> This was criticized by Jeffrey Oaks, who claims that the study of curves by means of equations originated with Descartes in the seventeenth century.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Renaissance">Renaissance</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=16" title="Edit section: Renaissance"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of <a href="/wiki/Renaissance" title="Renaissance">Renaissance</a> mathematicians such as <a href="/wiki/Gerolamo_Cardano" title="Gerolamo Cardano">Gerolamo Cardano</a> and <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" class="mw-redirect" title="Niccolò Fontana Tartaglia">Niccolò Fontana "Tartaglia"</a> on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was favored by most 16th and 17th century mathematicians, notably <a href="/wiki/Blaise_Pascal" title="Blaise Pascal">Blaise Pascal</a> who argued against the use of algebraic and analytical methods in geometry.<sup id="cite_ref-FOOTNOTEKline1972279_13-0" class="reference"><a href="#cite_note-FOOTNOTEKline1972279-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The French mathematicians <a href="/wiki/Franciscus_Vieta" class="mw-redirect" title="Franciscus Vieta">Franciscus Vieta</a> and later <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> and <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> revolutionized the conventional way of thinking about construction problems through the introduction of <a href="/wiki/Coordinate_geometry" class="mw-redirect" title="Coordinate geometry">coordinate geometry</a>. They were interested primarily in the properties of <i>algebraic curves</i>, such as those defined by <a href="/wiki/Diophantine_equations" class="mw-redirect" title="Diophantine equations">Diophantine equations</a> (in the case of Fermat), and the algebraic reformulation of the classical Greek works on conics and cubics (in the case of Descartes). </p><p>During the same period, Blaise Pascal and <a href="/wiki/G%C3%A9rard_Desargues" class="mw-redirect" title="Gérard Desargues">Gérard Desargues</a> approached geometry from a different perspective, developing the <a href="/wiki/Synthetic_geometry" title="Synthetic geometry">synthetic</a> notions of <a href="/wiki/Projective_geometry" title="Projective geometry">projective geometry</a>. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek <i>ruler and compass construction</i>. Ultimately, the <a href="/wiki/Analytic_geometry" title="Analytic geometry">analytic geometry</a> of Descartes and Fermat won out, for it supplied the 18th century mathematicians with concrete quantitative tools needed to study physical problems using the new calculus of <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a> and <a href="/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Leibniz</a>. However, by the end of the 18th century, most of the algebraic character of coordinate geometry was subsumed by the <i>calculus of infinitesimals</i> of <a href="/wiki/Joseph_Louis_Lagrange" class="mw-redirect" title="Joseph Louis Lagrange">Lagrange</a> and <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a>. </p> <div class="mw-heading mw-heading3"><h3 id="19th_and_early_20th_century">19th and early 20th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=17" title="Edit section: 19th and early 20th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It took the simultaneous 19th century developments of <a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">non-Euclidean geometry</a> and <a href="/wiki/Abelian_integral" title="Abelian integral">Abelian integrals</a> in order to bring the old algebraic ideas back into the geometrical fold. The first of these new developments was seized up by <a href="/wiki/Edmond_Laguerre" title="Edmond Laguerre">Edmond Laguerre</a> and <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a>, who attempted to ascertain the generalized metric properties of projective space. Cayley introduced the idea of <i>homogeneous polynomial forms</i>, and more specifically <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a>, on projective space. Subsequently, <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> studied projective geometry (along with other types of geometry) from the viewpoint that the geometry on a space is encoded in a certain class of <a href="/wiki/Transformation_group" class="mw-redirect" title="Transformation group">transformations</a> on the space. By the end of the 19th century, projective geometers were studying more general kinds of transformations on figures in projective space. Rather than the projective linear transformations which were normally regarded as giving the fundamental <a href="/wiki/Kleinian_geometry" class="mw-redirect" title="Kleinian geometry">Kleinian geometry</a> on projective space, they concerned themselves also with the higher degree <a href="/wiki/Birational_transformation" class="mw-redirect" title="Birational transformation">birational transformations</a>. This weaker notion of congruence would later lead members of the 20th century <a href="/wiki/Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry</a> to classify <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surfaces</a> up to <a href="/wiki/Birational_isomorphism" class="mw-redirect" title="Birational isomorphism">birational isomorphism</a>. </p><p>The second early 19th century development, that of Abelian integrals, would lead <a href="/wiki/Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a> to the development of <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>. </p><p>In the same period began the algebraization of the algebraic geometry through <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>. The prominent results in this direction are <a href="/wiki/Hilbert%27s_basis_theorem" title="Hilbert's basis theorem">Hilbert's basis theorem</a> and <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a>, which are the basis of the connection between algebraic geometry and commutative algebra, and <a href="/wiki/Francis_Sowerby_Macaulay" title="Francis Sowerby Macaulay">Macaulay</a>'s <a href="/wiki/Multivariate_resultant" class="mw-redirect" title="Multivariate resultant">multivariate resultant</a>, which is the basis of <a href="/wiki/Elimination_theory" title="Elimination theory">elimination theory</a>. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by <a href="/wiki/Singularity_theory" title="Singularity theory">singularity theory</a> and computational algebraic geometry.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="20th_century">20th century</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=18" title="Edit section: 20th century"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/B._L._van_der_Waerden" class="mw-redirect" title="B. L. van der Waerden">B. L. van der Waerden</a>, <a href="/wiki/Oscar_Zariski" title="Oscar Zariski">Oscar Zariski</a> and <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a> developed a foundation for algebraic geometry based on contemporary <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, including <a href="/wiki/Valuation_theory" class="mw-redirect" title="Valuation theory">valuation theory</a> and the theory of <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a>. One of the goals was to give a rigorous framework for proving the results of the <a href="/wiki/Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry</a>. In particular, this school used systematically the notion of <a href="/wiki/Generic_point" title="Generic point">generic point</a> without any precise definition, which was first given by these authors during the 1930s. </p><p>In the 1950s and 1960s, <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a> and <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> recast the foundations making use of <a href="/wiki/Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a>. Later, from about 1960, and largely led by Grothendieck, the idea of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> was worked out, in conjunction with a very refined apparatus of <a href="/wiki/Homological_algebra" title="Homological algebra">homological techniques</a>. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both to <a href="/wiki/Number_theory" title="Number theory">number theory</a> and to more classical geometric questions on algebraic varieties, <a href="/wiki/Singularity_theory" title="Singularity theory">singularities</a>, <a href="/wiki/Moduli_space" title="Moduli space">moduli</a>, and <a href="/wiki/Formal_moduli" title="Formal moduli">formal moduli</a>. </p><p>An important class of varieties, not easily understood directly from their defining equations, are the <a href="/wiki/Abelian_variety" title="Abelian variety">abelian varieties</a>, which are the projective varieties whose points form an abelian <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a>. The prototypical examples are the <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curves</a>, which have a rich theory. They were instrumental in the proof of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> and are also used in <a href="/wiki/Elliptic-curve_cryptography" title="Elliptic-curve cryptography">elliptic-curve cryptography</a>. </p><p>In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods for effective computation with concretely-given varieties have also been developed, which lead to the new area of computational algebraic geometry. One of the founding methods of this area is the theory of <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner bases</a>, introduced by <a href="/wiki/Bruno_Buchberger" title="Bruno Buchberger">Bruno Buchberger</a> in 1965. Another founding method, more specially devoted to real algebraic geometry, is the <a href="/wiki/Cylindrical_algebraic_decomposition" title="Cylindrical algebraic decomposition">cylindrical algebraic decomposition</a>, introduced by <a href="/wiki/George_E._Collins" title="George E. Collins">George E. Collins</a> in 1973. </p><p>See also: <a href="/wiki/Derived_algebraic_geometry" title="Derived algebraic geometry">derived algebraic geometry</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Analytic_geometry">Analytic geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=19" title="Edit section: Analytic geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <b><a href="/wiki/Analytic_variety" class="mw-redirect" title="Analytic variety">analytic variety</a></b> over the field of real or complex numbers is defined locally as the set of common solutions of several equations involving <a href="/wiki/Analytic_function" title="Analytic function">analytic functions</a>. It is analogous to the concept of <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> in that it carries a structure sheaf of analytic functions instead of regular functions. Any <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a> is a complex analytic variety. Since analytic varieties may have <a href="/wiki/Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singular points</a>, not all complex analytic varieties are manifolds. Over a non-archimedean field analytic geometry is studied via <a href="/wiki/Rigid_analytic_space" title="Rigid analytic space">rigid analytic spaces</a>. </p><p>Modern analytic geometry over the field of complex numbers is closely related to complex algebraic geometry, as has been shown by <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a> in his paper <i><a href="/wiki/GAGA" class="mw-redirect" title="GAGA">GAGA</a></i>,<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> the name of which is French for <i>Algebraic geometry and analytic geometry</i>. The GAGA results over the field of complex numbers may be extended to rigid analytic spaces over non-archimedean fields.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=20" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algebraic geometry now finds applications in <a href="/wiki/Algebraic_statistics" title="Algebraic statistics">statistics</a>,<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Control_theory" title="Control theory">control theory</a>,<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Robotics" title="Robotics">robotics</a>,<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Algebraic_geometric_code" class="mw-redirect" title="Algebraic geometric code">error-correcting codes</a>,<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Computational_phylogenetics" title="Computational phylogenetics">phylogenetics</a><sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Geometric_modelling" class="mw-redirect" title="Geometric modelling">geometric modelling</a>.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> There are also connections to <a href="/wiki/Homological_mirror_symmetry" title="Homological mirror symmetry">string theory</a>,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Game_theory" title="Game theory">game theory</a>,<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Matching_(graph_theory)" title="Matching (graph theory)">graph matchings</a>,<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Soliton" title="Soliton">solitons</a><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Integer_programming" title="Integer programming">integer programming</a>.<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 27em;"> <ul><li><a href="/wiki/Glossary_of_classical_algebraic_geometry" title="Glossary of classical algebraic geometry">Glossary of classical algebraic geometry</a></li> <li><a href="/wiki/List_of_publications_in_mathematics#Algebraic_geometry" class="mw-redirect" title="List of publications in mathematics">Important publications in algebraic geometry</a></li> <li><a href="/wiki/List_of_algebraic_surfaces" class="mw-redirect" title="List of algebraic surfaces">List of algebraic surfaces</a></li> <li><a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=22" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text">A witness of this oblivion is the fact that <a href="/wiki/Van_der_Waerden" class="mw-redirect" title="Van der Waerden">Van der Waerden</a> removed the chapter on elimination theory from the third edition (and all the subsequent ones) of his treatise <i>Moderne algebra</i> (in German).<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (January 2020)">citation needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.cs.sfu.ca/~ggbaker/zju/math/complexity.html">"Complexity of Algorithms"</a>. <i>www.cs.sfu.ca</i><span class="reference-accessdate">. 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"Dimers and Amoebae". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math-ph/0311005">math-ph/0311005</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Dimers+and+Amoebae&rft.date=2003&rft_id=info%3Aarxiv%2Fmath-ph%2F0311005&rft.aulast=Kenyon&rft.aufirst=Richard&rft.au=Okounkov%2C+Andrei&rft.au=Sheffield%2C+Scott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFordy1990" class="citation book cs1">Fordy, Allan P. (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=eO_PAAAAIAAJ"><i>Soliton Theory A Survey of Results</i></a>. Manchester University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7190-1491-8" title="Special:BookSources/978-0-7190-1491-8"><bdi>978-0-7190-1491-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Soliton+Theory+A+Survey+of+Results&rft.pub=Manchester+University+Press&rft.date=1990&rft.isbn=978-0-7190-1491-8&rft.aulast=Fordy&rft.aufirst=Allan+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DeO_PAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></span> </li> <li id="cite_note-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-28">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxSturmfels" class="citation book cs1"><a href="/wiki/David_A._Cox" title="David A. Cox">Cox, David A.</a>; Sturmfels, Bernd. Manocha, Dinesh N. (ed.). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=fe0MJEPDwzAC"><i>Applications of Computational Algebraic Geometry</i></a>. American Mathematical Soc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-6758-7" title="Special:BookSources/978-0-8218-6758-7"><bdi>978-0-8218-6758-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+of+Computational+Algebraic+Geometry&rft.pub=American+Mathematical+Soc.&rft.isbn=978-0-8218-6758-7&rft.aulast=Cox&rft.aufirst=David+A.&rft.au=Sturmfels%2C+Bernd&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dfe0MJEPDwzAC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=24" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1972" class="citation book cs1">Kline, M. (1972). <i>Mathematical Thought from Ancient to Modern Times</i>. Vol. 1. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0195061357" title="Special:BookSources/0195061357"><bdi>0195061357</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Thought+from+Ancient+to+Modern+Times&rft.pub=Oxford+University+Press&rft.date=1972&rft.isbn=0195061357&rft.aulast=Kline&rft.aufirst=M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=25" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dt>Some classic textbooks that predate schemes</dt> <dd></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Waerden1945" class="citation book cs1"><a href="/wiki/B._L._van_der_Waerden" class="mw-redirect" title="B. L. van der Waerden">van der Waerden, B. L.</a> (1945). <i>Einfuehrung in die algebraische Geometrie</i>. <a href="/wiki/Dover" title="Dover">Dover</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Einfuehrung+in+die+algebraische+Geometrie&rft.pub=Dover&rft.date=1945&rft.aulast=van+der+Waerden&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgePedoe1994" class="citation book cs1"><a href="/wiki/W._V._D._Hodge" title="W. V. D. Hodge">Hodge, W. V. D.</a>; <a href="/wiki/Daniel_Pedoe" title="Daniel Pedoe">Pedoe, Daniel</a> (1994). <i>Methods of Algebraic Geometry Volume 1</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46900-5" title="Special:BookSources/978-0-521-46900-5"><bdi>978-0-521-46900-5</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0796.14001">0796.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Algebraic+Geometry+Volume+1&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0796.14001%23id-name%3DZbl&rft.isbn=978-0-521-46900-5&rft.aulast=Hodge&rft.aufirst=W.+V.+D.&rft.au=Pedoe%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgePedoe1994" class="citation book cs1"><a href="/wiki/W._V._D._Hodge" title="W. V. D. Hodge">Hodge, W. V. D.</a>; <a href="/wiki/Daniel_Pedoe" title="Daniel Pedoe">Pedoe, Daniel</a> (1994). <i>Methods of Algebraic Geometry Volume 2</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46901-2" title="Special:BookSources/978-0-521-46901-2"><bdi>978-0-521-46901-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0796.14002">0796.14002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Algebraic+Geometry+Volume+2&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0796.14002%23id-name%3DZbl&rft.isbn=978-0-521-46901-2&rft.aulast=Hodge&rft.aufirst=W.+V.+D.&rft.au=Pedoe%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHodgePedoe1994" class="citation book cs1"><a href="/wiki/W._V._D._Hodge" title="W. V. D. Hodge">Hodge, W. V. D.</a>; <a href="/wiki/Daniel_Pedoe" title="Daniel Pedoe">Pedoe, Daniel</a> (1994). <i>Methods of Algebraic Geometry Volume 3</i>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-46775-9" title="Special:BookSources/978-0-521-46775-9"><bdi>978-0-521-46775-9</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0796.14003">0796.14003</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Methods+of+Algebraic+Geometry+Volume+3&rft.pub=Cambridge+University+Press&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0796.14003%23id-name%3DZbl&rft.isbn=978-0-521-46775-9&rft.aulast=Hodge&rft.aufirst=W.+V.+D.&rft.au=Pedoe%2C+Daniel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li></ul> <dl><dt>Modern textbooks that do not use the language of schemes</dt> <dd></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarrity2013" class="citation book cs1"><a href="/wiki/Thomas_A._Garrity" title="Thomas A. Garrity">Garrity, Thomas</a>; et al. (2013). <i>Algebraic Geometry A Problem Solving Approach</i>. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-821-89396-8" title="Special:BookSources/978-0-821-89396-8"><bdi>978-0-821-89396-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry+A+Problem+Solving+Approach&rft.pub=American+Mathematical+Society&rft.date=2013&rft.isbn=978-0-821-89396-8&rft.aulast=Garrity&rft.aufirst=Thomas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffithsHarris1994" class="citation book cs1"><a href="/wiki/Phillip_Griffiths" title="Phillip Griffiths">Griffiths, Phillip</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (1994). <i>Principles of Algebraic Geometry</i>. <a href="/wiki/Wiley-Interscience" class="mw-redirect" title="Wiley-Interscience">Wiley-Interscience</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-05059-9" title="Special:BookSources/978-0-471-05059-9"><bdi>978-0-471-05059-9</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0836.14001">0836.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Algebraic+Geometry&rft.pub=Wiley-Interscience&rft.date=1994&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0836.14001%23id-name%3DZbl&rft.isbn=978-0-471-05059-9&rft.aulast=Griffiths&rft.aufirst=Phillip&rft.au=Harris%2C+Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarris1995" class="citation book cs1"><a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (1995). <i>Algebraic Geometry A First Course</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97716-4" title="Special:BookSources/978-0-387-97716-4"><bdi>978-0-387-97716-4</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0779.14001">0779.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry+A+First+Course&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0779.14001%23id-name%3DZbl&rft.isbn=978-0-387-97716-4&rft.aulast=Harris&rft.aufirst=Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMumford1995" class="citation book cs1"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a> (1995). <i>Algebraic Geometry I Complex Projective Varieties</i> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-58657-9" title="Special:BookSources/978-3-540-58657-9"><bdi>978-3-540-58657-9</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0821.14001">0821.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry+I+Complex+Projective+Varieties&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0821.14001%23id-name%3DZbl&rft.isbn=978-3-540-58657-9&rft.aulast=Mumford&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFReid1988" class="citation book cs1"><a href="/wiki/Miles_Reid" title="Miles Reid">Reid, Miles</a> (1988). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/undergraduatealg0000reid"><i>Undergraduate Algebraic Geometry</i></a></span>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-35662-6" title="Special:BookSources/978-0-521-35662-6"><bdi>978-0-521-35662-6</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0701.14001">0701.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Undergraduate+Algebraic+Geometry&rft.pub=Cambridge+University+Press&rft.date=1988&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0701.14001%23id-name%3DZbl&rft.isbn=978-0-521-35662-6&rft.aulast=Reid&rft.aufirst=Miles&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fundergraduatealg0000reid&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShafarevich1995" class="citation book cs1"><a href="/wiki/Igor_Shafarevich" title="Igor Shafarevich">Shafarevich, Igor</a> (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/basicalgebraicge00irsh"><i>Basic Algebraic Geometry I Varieties in Projective Space</i></a></span> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-54812-8" title="Special:BookSources/978-0-387-54812-8"><bdi>978-0-387-54812-8</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0797.14001">0797.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Algebraic+Geometry+I+Varieties+in+Projective+Space&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0797.14001%23id-name%3DZbl&rft.isbn=978-0-387-54812-8&rft.aulast=Shafarevich&rft.aufirst=Igor&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbasicalgebraicge00irsh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li></ul> <dl><dt>Textbooks in computational algebraic geometry</dt></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxLittleO'Shea1997" class="citation book cs1"><a href="/wiki/David_A._Cox" title="David A. Cox">Cox, David A.</a>; Little, John; O'Shea, Donal (1997). <i>Ideals, Varieties, and Algorithms</i> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94680-1" title="Special:BookSources/978-0-387-94680-1"><bdi>978-0-387-94680-1</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0861.13012">0861.13012</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ideals%2C+Varieties%2C+and+Algorithms&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1997&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0861.13012%23id-name%3DZbl&rft.isbn=978-0-387-94680-1&rft.aulast=Cox&rft.aufirst=David+A.&rft.au=Little%2C+John&rft.au=O%27Shea%2C+Donal&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchenck2003" class="citation book cs1"><a href="/wiki/Hal_Schenck" title="Hal Schenck">Schenck, Hal</a> (2003). <a rel="nofollow" class="external text" href="https://www.cambridge.org/us/academic/subjects/mathematics/geometry-and-topology/computational-algebraic-geometry?format=HB&isbn=9780521829649"><i>Computational Algebraic Geometry</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Computational+Algebraic+Geometry&rft.pub=Cambridge+University+Press&rft.date=2003&rft.aulast=Schenck&rft.aufirst=Hal&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fus%2Facademic%2Fsubjects%2Fmathematics%2Fgeometry-and-topology%2Fcomputational-algebraic-geometry%3Fformat%3DHB%26isbn%3D9780521829649&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBasuPollackRoy2006" class="citation book cs1">Basu, Saugata; Pollack, Richard; Roy, Marie-Françoise (2006). <a rel="nofollow" class="external text" href="http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html"><i>Algorithms in real algebraic geometry</i></a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+in+real+algebraic+geometry&rft.pub=Springer-Verlag&rft.date=2006&rft.aulast=Basu&rft.aufirst=Saugata&rft.au=Pollack%2C+Richard&rft.au=Roy%2C+Marie-Fran%C3%A7oise&rft_id=http%3A%2F%2Fperso.univ-rennes1.fr%2Fmarie-francoise.roy%2Fbpr-ed2-posted1.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGonzález-VegaRecio1996" class="citation book cs1">González-Vega, Laureano; Recio, Tómas (1996). <i>Algorithms in algebraic geometry and applications</i>. Birkhaüser.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+in+algebraic+geometry+and+applications&rft.pub=Birkha%C3%BCser&rft.date=1996&rft.aulast=Gonz%C3%A1lez-Vega&rft.aufirst=Laureano&rft.au=Recio%2C+T%C3%B3mas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFElkadiMourrainPiene2006" class="citation book cs1">Elkadi, Mohamed; Mourrain, Bernard; Piene, Ragni, eds. (2006). <i>Algebraic geometry and geometric modeling</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+geometry+and+geometric+modeling&rft.pub=Springer-Verlag&rft.date=2006&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickensteinSchreyerSommese2008" class="citation book cs1"><a href="/wiki/Alicia_Dickenstein" title="Alicia Dickenstein">Dickenstein, Alicia</a>; Schreyer, Frank-Olaf; Sommese, Andrew J., eds. (2008). <i>Algorithms in Algebraic Geometry</i>. The IMA Volumes in Mathematics and its Applications. Vol. 146. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387751559" title="Special:BookSources/9780387751559"><bdi>9780387751559</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/2007938208">2007938208</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algorithms+in+Algebraic+Geometry&rft.series=The+IMA+Volumes+in+Mathematics+and+its+Applications&rft.pub=Springer&rft.date=2008&rft_id=info%3Alccn%2F2007938208&rft.isbn=9780387751559&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxLittleO'Shea1998" class="citation book cs1"><a href="/wiki/David_A._Cox" title="David A. Cox">Cox, David A.</a>; Little, John B.; O'Shea, Donal (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-978-1-4757-6911-1"><i>Using algebraic geometry</i></a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Using+algebraic+geometry&rft.pub=Springer-Verlag&rft.date=1998&rft.aulast=Cox&rft.aufirst=David+A.&rft.au=Little%2C+John+B.&rft.au=O%27Shea%2C+Donal&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspringer_10.1007-978-1-4757-6911-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCavinessJohnson1998" class="citation book cs1">Caviness, Bob F.; Johnson, Jeremy R. (1998). <i>Quantifier elimination and cylindrical algebraic decomposition</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantifier+elimination+and+cylindrical+algebraic+decomposition&rft.pub=Springer-Verlag&rft.date=1998&rft.aulast=Caviness&rft.aufirst=Bob+F.&rft.au=Johnson%2C+Jeremy+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li></ul> <dl><dt>Textbooks and references for schemes</dt> <dd></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisenbudHarris1998" class="citation book cs1"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, David</a>; <a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-978-0-387-22639-2"><i>The Geometry of Schemes</i></a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98637-1" title="Special:BookSources/978-0-387-98637-1"><bdi>978-0-387-98637-1</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0960.14002">0960.14002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Schemes&rft.pub=Springer-Verlag&rft.date=1998&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0960.14002%23id-name%3DZbl&rft.isbn=978-0-387-98637-1&rft.aulast=Eisenbud&rft.aufirst=David&rft.au=Harris%2C+Joe&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspringer_10.1007-978-0-387-22639-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrothendieck1960" class="citation book cs1"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexander</a> (1960). <a href="/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique" title="Éléments de géométrie algébrique"><i>Éléments de géométrie algébrique</i></a>. <a href="/wiki/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0118.36206">0118.36206</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89l%C3%A9ments+de+g%C3%A9om%C3%A9trie+alg%C3%A9brique&rft.pub=Publications+Math%C3%A9matiques+de+l%27IH%C3%89S&rft.date=1960&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0118.36206%23id-name%3DZbl&rft.aulast=Grothendieck&rft.aufirst=Alexander&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrothendieckDieudonné1971" class="citation book cs1"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexander</a>; Dieudonné, Jean Alexandre (1971). <a href="/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique" title="Éléments de géométrie algébrique"><i>Éléments de géométrie algébrique</i></a>. Vol. 1 (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-05113-8" title="Special:BookSources/978-3-540-05113-8"><bdi>978-3-540-05113-8</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0203.23301">0203.23301</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%C3%89l%C3%A9ments+de+g%C3%A9om%C3%A9trie+alg%C3%A9brique&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1971&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0203.23301%23id-name%3DZbl&rft.isbn=978-3-540-05113-8&rft.aulast=Grothendieck&rft.aufirst=Alexander&rft.au=Dieudonn%C3%A9%2C+Jean+Alexandre&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHartshorne1977" class="citation book cs1"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977). <a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)"><i>Algebraic Geometry</i></a>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90244-9" title="Special:BookSources/978-0-387-90244-9"><bdi>978-0-387-90244-9</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0367.14001">0367.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.pub=Springer-Verlag&rft.date=1977&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0367.14001%23id-name%3DZbl&rft.isbn=978-0-387-90244-9&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMumford1999" class="citation book cs1"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a> (1999). <i>The Red Book of Varieties and Schemes Includes the Michigan Lectures on Curves and Their Jacobians</i> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-63293-1" title="Special:BookSources/978-3-540-63293-1"><bdi>978-3-540-63293-1</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0945.14001">0945.14001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Red+Book+of+Varieties+and+Schemes+Includes+the+Michigan+Lectures+on+Curves+and+Their+Jacobians&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1999&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0945.14001%23id-name%3DZbl&rft.isbn=978-3-540-63293-1&rft.aulast=Mumford&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShafarevich1995" class="citation book cs1"><a href="/wiki/Igor_Shafarevich" title="Igor Shafarevich">Shafarevich, Igor</a> (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/basicalgebraicge00irsh"><i>Basic Algebraic Geometry II Schemes and complex manifolds</i></a></span> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-57554-2" title="Special:BookSources/978-3-540-57554-2"><bdi>978-3-540-57554-2</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0797.14002">0797.14002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Algebraic+Geometry+II+Schemes+and+complex+manifolds&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0797.14002%23id-name%3DZbl&rft.isbn=978-3-540-57554-2&rft.aulast=Shafarevich&rft.aufirst=Igor&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbasicalgebraicge00irsh&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_geometry&action=edit&section=26" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 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has quotations related to <i><b><a href="https://en.wikiquote.org/wiki/Special:Search/Algebraic_geometry" class="extiw" title="q:Special:Search/Algebraic geometry">Algebraic geometry</a></b></i>.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf"><i>Foundations of Algebraic Geometry</i> by Ravi Vakil, 808 pp.</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20040415021548/http://planetmath.org/encyclopedia/AlgebraicGeometry.html"><i>Algebraic geometry</i></a> entry on <a rel="nofollow" class="external text" href="https://planetmath.org/">PlanetMath</a></li> <li><a rel="nofollow" class="external text" href="http://neo-classical-physics.info/uploads/3/0/6/5/3065888/van_der_waerden_-_algebraic_geometry.pdf">English translation of the van der Waerden textbook</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDieudonné1972" class="citation web cs1"><a href="/wiki/Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (March 3, 1972). <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=Jzx-0poj3Eo">"The History of Algebraic Geometry"</a>. Talk at the Department of Mathematics of the <a href="/wiki/University_of_Wisconsin%E2%80%93Milwaukee" title="University of Wisconsin–Milwaukee">University of Wisconsin–Milwaukee</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/varchive/youtube/20211122/Jzx-0poj3Eo">Archived</a> from the original on 2021-11-22 – via <a href="/wiki/YouTube" title="YouTube">YouTube</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+History+of+Algebraic+Geometry&rft.pub=Talk+at+the+Department+of+Mathematics+of+the+University+of+Wisconsin%E2%80%93Milwaukee&rft.date=1972-03-03&rft.aulast=Dieudonn%C3%A9&rft.aufirst=Jean&rft_id=https%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DJzx-0poj3Eo&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+geometry" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/">The Stacks Project</a>, an open source textbook and reference work on algebraic stacks and algebraic geometry</li> <li><a rel="nofollow" class="external text" href="https://jessetvogel.nl/adjectives-project/">Adjectives Project</a>, an online database for searching examples of schemes and morphisms based on their properties</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output 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scope="row" class="navbox-group" style="width:1%">Areas</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract algebra</a></li> <li><a class="mw-selflink selflink">Algebraic geometry</a> <ul><li><a href="/wiki/Algebraic_variety" title="Algebraic variety">Algebraic variety</a></li> <li><a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">Scheme</a></li></ul></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary algebra</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li> <li><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li> <li><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_expression" title="Algebraic expression">Algebraic expression</a></li> <li><a href="/wiki/Equation" title="Equation">Equation</a> (<a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a>, <a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a>)</li> <li><a href="/wiki/Function_(mathematics)" title="Function (mathematics)">Function</a> (<a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">Polynomial function</a>)</li> <li><a href="/wiki/Inequality_(mathematics)" title="Inequality (mathematics)">Inequality</a> (<a href="/wiki/Linear_inequality" title="Linear inequality">Linear inequality</a>)</li> <li><a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">Operation</a> (<a href="/wiki/Addition" title="Addition">Addition</a>, <a href="/wiki/Multiplication" title="Multiplication">Multiplication</a>)</li> <li><a href="/wiki/Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="/wiki/Equivalence_relation" title="Equivalence relation">Equivalence relation</a>)</li> <li><a href="/wiki/Variable_(mathematics)" title="Variable (mathematics)">Variable</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic structures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> (<a href="/wiki/Field_theory_(mathematics)" class="mw-redirect" title="Field theory (mathematics)">theory</a>)</li> <li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a> (<a href="/wiki/Group_theory" title="Group theory">theory</a>)</li> <li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a> (<a href="/wiki/Commutative_algebra" title="Commutative algebra">theory</a>)</li> <li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a> (<a href="/wiki/Ring_theory" title="Ring theory">theory</a>)</li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a> (<a href="/wiki/Coordinate_vector" title="Coordinate vector">Vector</a>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Linear and <br />multilinear algebra</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a> (<a href="/wiki/Dot_product" title="Dot product">Dot product</a>)</li> <li><a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a> (<a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a>)</li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Linear subspace</a> (<a href="/wiki/Affine_space" title="Affine space">Affine space</a>)</li> <li><a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">Norm</a> (<a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>)</li> <li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a> (<a href="/wiki/Orthogonal_complement" title="Orthogonal complement">Orthogonal complement</a>)</li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">Trace</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Algebraic constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Free_object" title="Free object">Free object</a> (<a href="/wiki/Free_group" title="Free group">Free group</a>, ...)</li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a> (<a href="/wiki/Multivector" title="Multivector">Multivector</a>)</li> <li><a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a> (<a href="/wiki/Polynomial" title="Polynomial">Polynomial</a>)</li> <li><a href="/wiki/Quotient_object" class="mw-redirect" title="Quotient object">Quotient object</a> (<a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a>, ...)</li> <li><a href="/wiki/Symmetric_algebra" title="Symmetric algebra">Symmetric algebra</a></li> <li><a href="/wiki/Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Topic lists</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category:Algebraic_structures" title="Category:Algebraic structures">Algebraic structures</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Glossaries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Glossary_of_field_theory" title="Glossary of field theory">Field theory</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Linear algebra</a></li> <li><a href="/wiki/Glossary_of_order_theory" title="Glossary of order theory">Order theory</a></li> <li><a href="/wiki/Glossary_of_ring_theory" title="Glossary of ring theory">Ring theory</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Algebra" title="Category:Algebra">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Geometry" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Geometry" title="Template:Geometry"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Geometry" title="Template talk:Geometry"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geometry" title="Special:EditPage/Template:Geometry"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Geometry" style="font-size:114%;margin:0 4em"><a href="/wiki/Geometry" title="Geometry">Geometry</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/History_of_geometry" title="History of geometry">History</a> <ul><li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">Timeline</a></li></ul></li> <li><a href="/wiki/Lists_of_geometry_topics" class="mw-redirect" title="Lists of geometry topics">Lists</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean <br /> geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Combinatorial</a></li> <li><a href="/wiki/Convex_geometry" title="Convex geometry">Convex</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">Plane geometry</a> <ul><li><a href="/wiki/Polygon" title="Polygon">Polygon</a></li> <li><a href="/wiki/Polyform" title="Polyform">Polyform</a></li></ul></li> <li><a href="/wiki/Solid_geometry" title="Solid geometry">Solid geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Non-Euclidean_geometry" title="Non-Euclidean geometry">Non-Euclidean <br /> geometry</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_geometry" title="Elliptic geometry">Elliptic</a></li> <li><a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">Hyperbolic</a></li> <li><a href="/wiki/Symplectic_geometry" title="Symplectic geometry">Symplectic</a></li> <li><a href="/wiki/Spherical_geometry" title="Spherical geometry">Spherical</a></li> <li><a href="/wiki/Affine_geometry" title="Affine geometry">Affine</a></li> <li><a href="/wiki/Projective_geometry" title="Projective geometry">Projective</a></li> <li><a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Trigonometry" title="Trigonometry">Trigonometry</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li> <li><a class="mw-selflink selflink">Algebraic geometry</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Shape" title="Shape">Shape</a> <ul><li><a href="/wiki/Lists_of_shapes" title="Lists of shapes">Lists</a></li></ul></li> <li><a href="/wiki/List_of_geometry_topics" class="mw-redirect" title="List of geometry topics">List of geometry topics</a></li> <li><a href="/wiki/List_of_differential_geometry_topics" title="List of differential geometry topics">List of differential geometry topics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Geometry" title="Category:Geometry">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="Polynomials_and_polynomial_functions" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Polynomials" title="Template:Polynomials"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Polynomials" title="Template talk:Polynomials"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Polynomials" title="Special:EditPage/Template:Polynomials"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Polynomials_and_polynomial_functions" style="font-size:114%;margin:0 4em"><a href="/wiki/Polynomial" title="Polynomial">Polynomials</a> and <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Zero_polynomial" class="mw-redirect" title="Zero polynomial">Zero polynomial (degree undefined or −1 or −∞)</a></li> <li><a href="/wiki/Constant_function" title="Constant function">Constant function (0)</a></li> <li><a href="/wiki/Linear_function_(calculus)" title="Linear function (calculus)">Linear function (1)</a> <ul><li><a href="/wiki/Linear_equation" title="Linear equation">Linear equation</a></li></ul></li> <li><a href="/wiki/Quadratic_function" title="Quadratic function">Quadratic function (2)</a> <ul><li><a href="/wiki/Quadratic_equation" title="Quadratic equation">Quadratic equation</a></li></ul></li> <li><a href="/wiki/Cubic_function" title="Cubic function">Cubic function (3)</a> <ul><li><a href="/wiki/Cubic_equation" title="Cubic equation">Cubic equation</a></li></ul></li> <li><a href="/wiki/Quartic_function" title="Quartic function">Quartic function (4)</a> <ul><li><a href="/wiki/Quartic_equation" title="Quartic equation">Quartic equation</a></li></ul></li> <li><a href="/wiki/Quintic_function" title="Quintic function">Quintic function (5)</a></li> <li><a href="/wiki/Sextic_equation" title="Sextic equation">Sextic equation (6)</a></li> <li><a href="/wiki/Septic_equation" title="Septic equation">Septic equation (7)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By properties</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Univariate_polynomial" class="mw-redirect" title="Univariate polynomial">Univariate</a></li> <li><a href="/wiki/Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">Bivariate</a></li> <li><a href="/wiki/Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">Multivariate</a></li> <li><a href="/wiki/Monomial" title="Monomial">Monomial</a></li> <li><a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">Binomial</a></li> <li><a href="/wiki/Trinomial" title="Trinomial">Trinomial</a></li> <li><a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">Irreducible</a></li> <li><a href="/wiki/Square-free_polynomial" title="Square-free polynomial">Square-free</a></li> <li><a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">Homogeneous</a></li> <li><a href="/wiki/Quasi-homogeneous_polynomial" title="Quasi-homogeneous polynomial">Quasi-homogeneous</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools and algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Factorization_of_polynomials" title="Factorization of polynomials">Factorization</a></li> <li><a href="/wiki/Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">Greatest common divisor</a></li> <li><a href="/wiki/Polynomial_long_division" title="Polynomial long division">Division</a></li> <li><a href="/wiki/Horner%27s_method" title="Horner's method">Horner's method of evaluation</a></li> <li><a href="/wiki/Polynomial_identity_testing" title="Polynomial identity testing">Polynomial identity testing</a></li> <li><a href="/wiki/Polynomial_resultant" class="mw-redirect" title="Polynomial resultant">Resultant</a></li> <li><a href="/wiki/Discriminant" title="Discriminant">Discriminant</a></li> <li><a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></li></ul> </div></td></tr></tbody></table></div> <p><br /> </p> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1038841319"><link 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