%!PS-Adobe-2.0 %%Creator: dvips(k) 2023.1 (TeX Live 2023) Copyright 2023 Radical Eye Software %%Title: arXiv:2412.14420v1 [math.CO] 19 Dec 2024 %%CreationDate: Tue Jan 14 13:38:35 2025 %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z main.dvi -o %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2025.01.14:1338 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: l3backend-dvips.pro 0 0 %% %% This is file `l3backend-dvips.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% l3backend-header.dtx (with options: `header,dvips') %% %% Copyright (C) 2019-2023 The LaTeX Project %% %% It may be distributed and/or modified under the conditions of %% the LaTeX Project Public License (LPPL), either version 1.3c of %% this license or (at your option) any later version. 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b(claim)g(that)f(for)g(an)n(y)g(in)n (teger)g Fq(m;)14 b(n)37 b Fs(with)h Fm(j)p Fq(m)p Fm(j)p Fq(;)14 b Fm(j)p Fq(n)p Fm(j)39 b(\024)g Fs(5)p Fq(N)t(;)f(m)25 b Fm(\000)f Fq(nt)40 b Fm(6)p Fs(=)f(0)e(in)150 2040 y Fp(F)195 2052 y Fo(p)239 2040 y Fq(:)c Fs(Supp)r(ose)g(for)f(con)n (tradiction)g(there)h(exists)f Fm(j)p Fq(m)1817 2052 y Fn(0)1854 2040 y Fm(j)p Fq(;)14 b Fm(j)p Fq(n)1987 2052 y Fn(0)2025 2040 y Fm(j)32 b(\024)f Fs(5)p Fq(N)41 b Fs(so)33 b(that)g Fq(m)2692 2052 y Fn(0)2751 2040 y Fm(\000)21 b Fq(n)2887 2052 y Fn(0)2924 2040 y Fq(t)32 b Fs(=)g(0)p Fq(:)g Fs(Without)i(loss)e(of)150 2140 y(generalit)n(y)26 b(w)n(e)h(ma)n(y)g(assume)g(gcd\()p Fq(m)1352 2152 y Fn(0)1389 2140 y Fq(;)14 b(n)1476 2152 y Fn(0)1513 2140 y Fs(\))23 b(=)g(1)p Fq(:)k Fs(Then)h(\()1997 2140 y SDict begin H.S end 1997 2140 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 2039 2086 a SDict begin H.R end 2039 2086 a 2039 2140 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.1) cvn H.B /ANN pdfmark end 2039 2140 a Fs(\))g(implies)1586 2325 y(\()1628 2269 y Fq(m)1701 2281 y Fn(0)p 1628 2306 111 4 v 1640 2382 a Fq(n)1690 2394 y Fn(0)1748 2325 y Fs(\))1780 2291 y Fn(2)1841 2325 y Fs(=)23 b Fq(a)1973 2337 y Fn(0)2028 2325 y Fs(+)2121 2269 y Fq(b)2157 2281 y Fn(0)2194 2269 y Fq(m)2267 2281 y Fn(0)p 2121 2306 184 4 v 2169 2382 a Fq(n)2219 2394 y Fn(0)150 2503 y Fs(and)k(hence)150 2645 y(\(2\))1451 2645 y SDict begin H.S end 1451 2645 a 1451 2645 a SDict begin 12 H.A end 1451 2645 a 1451 2645 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.2) cvn /DEST pdfmark end 1451 2645 a Fq(m)1524 2610 y Fn(2)1524 2665 y(0)1585 2645 y Fs(=)22 b Fq(a)1716 2657 y Fn(0)1753 2645 y Fq(n)1803 2610 y Fn(2)1803 2665 y(0)1859 2645 y Fs(+)c Fq(b)1978 2657 y Fn(0)2015 2645 y Fq(m)2088 2657 y Fn(0)2125 2645 y Fq(n)2175 2657 y Fn(0)2240 2645 y Fs(in)27 b Fp(F)2381 2657 y Fo(p)2426 2645 y Fq(:)150 2793 y Fs(By)g(assumption)g Fm(j)p Fq(B)t Fm(j)d Fs(=)e(\(2)p Fq(N)27 b Fs(+)18 b(1\))1268 2763 y Fn(2)1328 2793 y Fs(=)23 b Fq(o)p Fs(\()p Fq(p)1530 2763 y Fn(2)p Fo(=)p Fn(3)1635 2793 y Fs(\))p Fq(;)28 b Fs(when)g Fq(p)f Fs(is)g(su\016cien)n(tly)h(large)e(w)n(e)h(get)1423 2956 y Fq(m)1496 2922 y Fn(2)1496 2977 y(0)1556 2956 y Fq(<)1654 2900 y(p)p 1654 2937 42 4 v 1654 3013 a Fs(2)1706 2956 y Fq(;)41 b Fm(j)p Fq(an)1887 2922 y Fn(2)1887 2977 y(0)1943 2956 y Fs(+)18 b Fq(bm)2135 2968 y Fn(0)2171 2956 y Fq(n)2221 2968 y Fn(0)2258 2956 y Fm(j)24 b Fq(<)2402 2900 y(p)p 2402 2937 V 2402 3013 a Fs(2)2454 2956 y Fq(:)150 3131 y Fs(Th)n(us)34 b(\()399 3131 y SDict begin H.S end 399 3131 a 0 TeXcolorgray Fs(2)p 0 TeXcolorgray 441 3078 a SDict begin H.R end 441 3078 a 441 3131 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 441 3131 a Fs(\))h(holds)f(not)g(only)g(in)h Fp(F)1224 3143 y Fo(p)1303 3131 y Fs(but)g(also)e(in)i Fp(Z)o Fq(:)29 b Fs(This)34 b(implies)h Fq(n)2387 3143 y Fn(0)2459 3131 y Fm(j)f Fq(m)2589 3101 y Fn(2)2589 3152 y(0)2661 3131 y Fs(and)g(hence)h Fm(j)p Fq(n)3140 3143 y Fn(0)3177 3131 y Fm(j)f Fs(=)g(1)p Fq(;)14 b(m)3485 3143 y Fn(0)3556 3131 y Fm(\021)34 b(\006)p Fq(t)150 3231 y Fs(\(mo)r(d)28 b Fq(p)p Fs(\).)37 b(F)-7 b(rom)27 b(the)h(fact)g(that)g Fq(B)k Fs(is)27 b(prop)r(er)g(w)n(e)g(also)g(kno)n(w)f Fm(j)p Fq(m)2253 3243 y Fn(0)2291 3231 y Fm(j)d(\025)f(j)p Fq(t)p Fm(j)h(\025)g Fq(N)k Fs(+)18 b(1)p Fq(;)28 b Fs(therefore)1481 3374 y Fq(m)1554 3340 y Fn(2)1554 3395 y(0)1609 3374 y Fm(\006)18 b Fq(b)1728 3386 y Fn(0)1765 3374 y Fq(m)1838 3386 y Fn(0)1898 3374 y Fm(\025)23 b Fq(N)k Fs(+)18 b(1)23 b Fq(>)f(a)2359 3386 y Fn(0)2396 3374 y Fq(;)150 3516 y Fs(whic)n(h)28 b(is)f(a)g(con)n(tradiction)f(to)i(\()1179 3516 y SDict begin H.S end 1179 3516 a 0 TeXcolorgray Fs(2)p 0 TeXcolorgray 1221 3462 a SDict begin H.R end 1221 3462 a 1221 3516 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 1221 3516 a Fs(\).)37 b(This)28 b(\014nishes)f(the)h(pro)r(of)f(of)h(the)g(claim.)250 3619 y(No)n(w)f(since)g Fq(\025t)720 3589 y Fn(2)781 3619 y Fm(2)c Fq(B)32 b Fs(for)27 b(an)n(y)g Fq(\025)c Fm(2)h Fs(\005)f(:=)g Fm(f)p Fq(uv)53 b Fs(:)e Fm(j)p Fq(u)p Fm(j)p Fq(;)14 b Fm(j)p Fq(v)s Fm(j)23 b(\024)f Fq(cN)2283 3589 y Fn(1)p Fo(=)p Fn(2)2387 3619 y Fm(g)p Fq(;)28 b Fs(consider)1547 3762 y Fm(f)p Fs(\()p Fq(\025)19 b Fs(+)f Fq(\026)p Fs(\))p Fq(t)1883 3728 y Fn(2)1943 3762 y Fs(:)23 b Fq(\025;)14 b(\026)24 b Fm(2)f Fs(\005)p Fm(g)p Fq(:)150 3907 y Fs(This)36 b(set)g(will)h(include)f Fq(w)r(t)1037 3877 y Fn(2)1112 3907 y Fs(for)f(an)n(y)h(in)n(teger)f Fm(j)p Fq(w)r Fm(j)j Fq(<)f(c)1979 3877 y Fn(2)2016 3907 y Fq(N)r(=)p Fs(2)p Fq(:)e Fs(\(T)-7 b(o)36 b(see)g(this,)j(tak)n(e)c Fq(b)i Fs(=)g Fm(b)p Fq(cN)3241 3877 y Fn(1)p Fo(=)p Fn(2)3345 3907 y Fm(c)g Fs(and)f(then)150 4007 y(consider)30 b(the)h(base-)p Fq(b)e Fs(expansion)h(of)h(all)f(p)r(ositiv)n(e)h(in)n (tegers)e Fm(\024)f Fq(c)2229 3976 y Fn(2)2266 4007 y Fq(N)r(=)p Fs(2,)j(and)g(note)f(that)h(suc)n(h)g(in)n(tegers)e(can)i(b) r(e)150 4109 y(written)d(as)f Fq(ab)18 b Fs(+)g(1)g Fm(\001)g Fq(r)r Fs(,)29 b(where)e(0)22 b Fm(\024)h Fq(a)g Fm(\024)g Fq(cN)1575 4079 y Fn(1)p Fo(=)p Fn(2)1706 4109 y Fs(and)28 b(0)22 b Fm(\024)h Fq(r)j Fm(\024)c Fq(cN)2282 4079 y Fn(1)p Fo(=)p Fn(2)2386 4109 y Fs(.\))38 b(Supp)r(ose)27 b(max\()p Fm(j)p Fq(a)3058 4121 y Fn(0)3095 4109 y Fm(j)p Fq(;)14 b Fm(j)p Fq(b)3214 4121 y Fn(0)3251 4109 y Fm(j)p Fs(\))24 b Fm(\025)3442 4077 y Fn(8)p 3427 4091 63 4 v 3427 4138 a Fo(c)3457 4121 y Fi(2)3499 4109 y Fq(;)k Fs(let)1471 4309 y Fq(w)e Fs(=)1644 4192 y Fl(\026)1823 4253 y Fs(2)p Fq(N)h Fs(+)18 b(1)p 1702 4290 503 4 v 1702 4366 a(max\()p Fm(j)p Fq(a)1956 4378 y Fn(0)1993 4366 y Fm(j)p Fq(;)c Fm(j)p Fq(b)2112 4378 y Fn(0)2149 4366 y Fm(j)p Fs(\))2214 4192 y Fl(\027)2281 4309 y Fs(+)k(1)p Fq(;)150 4506 y Fs(it)35 b(is)f(easy)f(to)h(c)n(hec)n(k)f(that)i Fq(w)i(<)c(c)1274 4476 y Fn(2)1311 4506 y Fq(N)r(=)p Fs(2)g(when)i Fq(p)f Fs(is)g(large,)g(hence)h Fq(w)r(t)2450 4476 y Fn(2)2522 4506 y Fm(2)f Fs(2)p Fq(B)t(:)g Fs(On)g(the)h(other)e (hand,)j(2)p Fq(N)42 b(<)150 4606 y Fs(max\()p Fm(j)p Fq(w)r(a)465 4618 y Fn(0)503 4606 y Fm(j)p Fq(;)14 b Fm(j)p Fq(w)r(b)683 4618 y Fn(0)720 4606 y Fm(j)p Fs(\))24 b Fm(\024)e Fs(3)p Fq(N)9 b Fs(.)37 b(F)-7 b(rom)27 b(the)h(claim)f(w)n (e)g(kno)n(w)1509 4749 y Fq(w)r(t)1600 4714 y Fn(2)1661 4749 y Fs(=)c Fq(w)r(a)1854 4761 y Fn(0)1910 4749 y Fs(+)18 b Fq(w)r(b)2090 4761 y Fn(0)2128 4749 y Fq(t)32 b(=)-51 b Fm(2)23 b Fs(2)p Fq(B)t(;)150 4891 y Fs(a)k(con)n(tradiction.)36 b(Hence)28 b(max)o(\()p Fm(j)p Fq(a)1257 4903 y Fn(0)1295 4891 y Fm(j)p Fq(;)14 b Fm(j)p Fq(b)1414 4903 y Fn(0)1450 4891 y Fm(j)p Fs(\))24 b Fm(\024)1641 4858 y Fn(8)p 1626 4872 63 4 v 1626 4919 a Fo(c)1656 4903 y Fi(2)1726 4891 y Fs(and)k(equation)f(\()2259 4891 y SDict begin H.S end 2259 4891 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 2300 4837 a SDict begin H.R end 2300 4837 a 2300 4891 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.1) cvn H.B /ANN pdfmark end 2300 4891 a Fs(\))h(giv)n(es)f(the)h(p)r(olynomial)f(w)n(e)g(w)n(an)n (t.)228 b Fh(\003)250 5051 y Fs(The)24 b(pro)r(of)g(strongly)e(dep)r (ends)j(on)f(the)h(prop)r(ert)n(y)e(that)h(enlarging)f Fq(B)28 b Fs(b)n(y)c(some)g(constan)n(t)f(will)i(still)f(giv)n(e)f(us)h (a)150 5150 y(prop)r(er)g(GAP)-7 b(.)25 b(W)-7 b(e)26 b(do)f(not)g(kno)n(w)f(if)i(it)f(remains)g(true)g(for)f Fm(j)p Fq(B)t Fm(j)f(\025)g Fq(p)2276 5120 y Fn(2)p Fo(=)p Fn(3)2380 5150 y Fs(.)36 b(This)25 b(motiv)-5 b(ates)25 b(us)g(to)g(de\014ne)h(a)e(sp)r(ecial)150 5250 y(class)j(of)g(GAPs.)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 142 -41 a SDict begin H.S end 142 -41 a 142 -41 a SDict begin H.R end 142 -41 a 142 -41 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end 142 -41 a 0 TeXcolorgray 150 34 a Fn(4)1210 b(ERNIE)21 b(CR)n(OOT)j(AND)e(JUNZHE)g(MA)n(O)p 0 TeXcolorgray 150 133 a SDict begin H.S end 150 133 a 150 133 a SDict begin 12 H.A end 150 133 a 150 133 a SDict begin [/View [/XYZ H.V]/Dest (defn.1) cvn /DEST pdfmark end 150 133 a 0 TeXcolorgray 100 x Fx(De\014nition)28 b(1.)p 0 TeXcolorgray 39 w Fj(A)g(gener)l(alize)l(d)h(arithmetic)g(pr)l(o)l(gr)l(ession)f Fq(B)g Fs(=)22 b Fq(x)2267 245 y Fn(0)2319 233 y Fs(+)14 b Fm(f)2440 171 y Fl(P)2527 191 y Fo(d)2527 258 y(i)p Fn(=1)2652 233 y Fq(a)2696 245 y Fo(i)2724 233 y Fq(x)2771 245 y Fo(i)2822 233 y Fs(:)23 b Fq(a)2912 245 y Fo(i)2963 233 y Fm(2)g Fs([0)p Fq(;)14 b(N)3210 245 y Fo(i)3251 233 y Fm(\000)g Fs(1])p Fm(g)26 b Fj(is)j(c)l(al)t(le)l(d)150 337 y Fq(\024)p Fj(-isolate)l(d)39 b(if)h(ther)l(e)e(is)h(no)g(nonzer)l (o)f Fs(\()p Fq(a)1461 349 y Fn(1)1499 337 y Fq(;)14 b(:::;)g(a)1686 349 y Fo(d)1724 337 y Fs(\))39 b Fm(2)g Fp(Z)1951 307 y Fo(d)2022 337 y Fj(with)g Fm(j)p Fq(a)2278 349 y Fo(i)2306 337 y Fm(j)f(\024)h Fq(\024N)2586 349 y Fo(i)2613 337 y Fj(,)i(for)e(al)t(l)h Fq(i)e Fs(=)h(1)p Fq(;)14 b(:::;)g(d)p Fj(,)40 b(such)f(that)150 374 y Fl(P)238 461 y Fo(i)279 436 y Fq(a)323 448 y Fo(i)351 436 y Fq(x)398 448 y Fo(i)449 436 y Fs(=)22 b(0)p Fq(:)250 588 y Fs(W)-7 b(orking)26 b(with)i(isolated)e(GAPs)h(w)n(e)g(are)f (able)h(to)g(generalize)f(Theorem)h(2)f(to)i(arbitrary)d(rank)h(and)h (pro)r(duct)150 687 y(set)h(of)f(t)n(w)n(o)g(di\013eren)n(t)h(GAPs.)36 b(Our)27 b(main)h(result)f(is)h(the)g(follo)n(wing:)150 739 y SDict begin H.S end 150 739 a 150 739 a SDict begin 12 H.A end 150 739 a 150 739 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3) cvn /DEST pdfmark end 150 739 a 0 TeXcolorgray 100 x Fx(Theorem)i(3.)p 0 TeXcolorgray 41 w Fj(L)l(et)f Fs(0)23 b Fq(<)g(c;)14 b(c)1076 809 y Fk(0)1099 839 y Fq(;)g(\017)22 b(<)h Fs(1)p Fq(;)14 b(d)23 b Fm(2)g Fp(N)t Fq(;)14 b(N)38 b Fs(=)22 b Fq(N)9 b Fs(\()p Fq(p)p Fs(\))24 b Fm(2)f Fp(N)40 b Fj(with)30 b Fs(lim)2455 851 y Fo(p)p Fk(!1)2640 839 y Fq(N)9 b Fs(\()p Fq(p)p Fs(\))23 b(=)g Fm(1)p Fj(,)1393 1061 y Fq(B)k Fs(=)c Fm(f)1655 958 y Fo(d)1613 983 y Fl(X)1619 1159 y Fo(i)p Fn(=1)1746 1061 y Fq(a)1790 1073 y Fo(i)1817 1061 y Fq(x)1864 1073 y Fo(i)1916 1061 y Fs(:)g Fm(j)p Fq(a)2029 1073 y Fo(i)2056 1061 y Fm(j)g(\024)g Fq(N)9 b Fm(g)22 b(\032)h Fp(F)2463 1073 y Fo(p)150 1272 y Fj(b)l(e)30 b Fs(6)p Fj(-isolate)l(d)g(with)h Fq(x)850 1284 y Fn(1)910 1272 y Fs(=)23 b(1)p Fq(:)29 b Fj(Supp)l(ose)h(ther)l(e)g(exist)g(pr)l (op)l(er)g(GAPs)880 1487 y Fq(A)24 b Fs(=)e Fm(f)1137 1383 y Fo(d)1095 1408 y Fl(X)1101 1585 y Fo(i)p Fn(=1)1228 1487 y Fq(a)1272 1499 y Fo(i)1300 1487 y Fq(y)1341 1499 y Fo(i)1391 1487 y Fs(:)i Fm(j)p Fq(a)1505 1499 y Fo(i)1532 1487 y Fm(j)f(\024)g 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Fs(=)22 b(\()p Fq(x)1425 3982 y Fn(1)1463 3970 y Fq(;)14 b(x)1547 3982 y Fn(2)1585 3970 y Fq(;)g(:)g(:)g(:)g(;)g(x)1817 3982 y Fo(d)1856 3970 y Fs(\))1888 3940 y Fo(T)1940 3970 y Fs(,)1996 3950 y Fq(~)1988 3970 y(u)2036 3946 y Fk(0)2082 3970 y Fs(=)22 b(\()p Fq(y)2245 3940 y Fk(0)2242 3991 y Fn(1)2280 3970 y Fq(;)14 b(y)2361 3940 y Fk(0)2358 3991 y Fn(2)2394 3970 y Fq(;)g(:)g(:)g(:)g(;)g(y)2623 3940 y Fk(0)2620 3994 y Fo(d)2658 3970 y Fs(\))2690 3940 y Fo(T)2766 3970 y Fm(2)23 b Fp(F)2889 3940 y Fo(d)2889 3991 y(p)2934 3970 y Fs(.)35 b(Also,)25 b(let)f Fq(T)34 b Fm(2)23 b Fq(M)3558 3982 y Fo(d)3596 3970 y Fs(\()p Fp(F)3673 3982 y Fo(p)3718 3970 y Fs(\))150 4070 y(b)r(e)28 b(the)g(matrix)f(whose)g(en)n(try)g(in)h(the)g Fq(j)5 b Fs(th)28 b(ro)n(w)e(and)i Fq(k)s Fs(th)g(column)f(is)1730 4231 y Fq(T)1779 4243 y Fo(j;k)1918 4231 y Fs(=)50 b Fq(a)2077 4188 y Fn(\()p Fo(k)q Fn(\))2077 4254 y Fo(ij)150 4386 y Fs(so)27 b(that)h(\()464 4386 y SDict begin H.S end 464 4386 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 506 4333 a SDict begin H.R end 506 4333 a 506 4386 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 506 4386 a Fs(\))g(b)r(ecomes)150 4545 y(\(5\))1734 4545 y SDict begin H.S end 1734 4545 a 1734 4545 a SDict begin 12 H.A end 1734 4545 a 1734 4545 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.5) cvn /DEST pdfmark end 1734 4545 a Fq(T)8 b(~)-38 b(v)53 b Fs(=)e Fq(y)2045 4557 y Fo(i)2080 4525 y Fq(~)2072 4545 y(u)2120 4521 y Fk(0)2143 4545 y Fq(:)150 4695 y Fs(F)-7 b(rom)27 b(\()400 4695 y SDict begin H.S end 400 4695 a 0 TeXcolorgray Fs(4)p 0 TeXcolorgray 442 4641 a SDict begin H.R end 442 4641 a 442 4695 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 442 4695 a Fs(\))h(it)g(follo)n(ws)f (that)g(all)h(the)g(en)n(tries)f(of)g(the)h(adjugate)f(matrix)g(adj\()p Fq(T)12 b Fs(\))27 b(are)g(b)r(ounded)h(b)n(y)1477 4848 y Fq(C)h Fs(=)23 b(\()p Fq(d)c Fm(\000)f Fs(1\)!3)1969 4814 y Fo(d)p Fk(\000)p Fn(1)2092 4848 y Fq(=)p Fs(\()p Fq(c)2202 4814 y Fk(00)2244 4848 y Fs(\))2276 4814 y Fo(d)p Fk(\000)p Fn(1)2400 4848 y Fq(:)150 4998 y Fs(Moreo)n(v)n(er,)46 b Fq(T)55 b Fs(m)n(ust)44 b(b)r(e)g(in)n(v)n(ertible)g(since)g (otherwise)f(from)h(Lemma)f(4,)48 b(there)c(exists)g(a)g(nonzero)e(v)n (ector)150 5098 y(\()p Fq(a)226 5110 y Fn(1)263 5098 y Fq(;)14 b(a)344 5110 y Fn(2)381 5098 y Fq(;)g(:)g(:)g(:)g(;)g(a)610 5110 y Fo(d)649 5098 y Fs(\))23 b Fm(2)g Fp(F)827 5067 y Fo(d)827 5118 y(p)899 5098 y Fs(with)29 b Fm(j)p Fq(a)1156 5110 y Fo(i)1183 5098 y Fm(j)f Fs(b)r(ounded)g(b)n(y)f(some)g(constan)n (t)g(dep)r(ending)h(on)f Fq(c)2781 5067 y Fk(00)2851 5098 y Fs(and)1567 5250 y(\()p Fq(a)1643 5262 y Fn(1)1680 5250 y Fq(;)14 b(a)1761 5262 y Fn(2)1798 5250 y Fq(;)g(:)g(:)g(:)g(;)g (a)2027 5262 y Fo(d)2065 5250 y Fs(\))p Fq(T)35 b Fs(=)22 b(0)p Fq(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 6 6 TeXDict begin HPSdict begin 6 5 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a 142 -41 a SDict begin H.S end 142 -41 a 142 -41 a SDict begin H.R end 142 -41 a 142 -41 a SDict begin [/View [/XYZ H.V]/Dest (page.6) cvn /DEST pdfmark end 142 -41 a 0 TeXcolorgray 150 34 a Fn(6)1210 b(ERNIE)21 b(CR)n(OOT)j(AND)e(JUNZHE)g(MA)n(O)p 0 TeXcolorgray 150 233 a Fs(Com)n(bining)27 b(this)h(with)g(\()955 233 y SDict begin H.S end 955 233 a 0 TeXcolorgray Fs(5)p 0 TeXcolorgray 997 180 a SDict begin H.R end 997 180 a 997 233 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.5) cvn H.B /ANN pdfmark end 997 233 a Fs(\))g(giv)n(es)e(us)1282 463 y Fq(y)1323 475 y Fo(i)1368 463 y Fm(\001)1453 359 y Fo(d)1410 384 y Fl(X)1412 561 y Fo(j)s 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1105 y Fs(T)-7 b(aking)27 b(linear)g(com)n(binations)f(of)i(co)r(ordinates)e(on)h(b)r(oth)h (sides)f(\(using)h(co)r(e\016cien)n(ts)f Fq(b)2915 1117 y Fo(j)2950 1105 y Fs(\),)h(w)n(e)f(can)g(deduce)1103 1304 y Fq(A)1165 1316 y Fn(1)1226 1304 y Fs(:=)22 b Fm(f)p Fs(det\()p Fq(T)12 b Fs(\))1632 1225 y Fl(X)1676 1402 y Fo(j)1765 1304 y Fq(b)1801 1316 y Fo(j)1836 1304 y Fq(x)1883 1316 y Fo(j)1941 1304 y Fs(:)24 b Fm(j)p Fq(b)2047 1316 y Fo(j)2081 1304 y Fm(j)f Fq(<)2228 1247 y(c)2264 1217 y Fk(0)p 2225 1285 66 4 v 2225 1361 a Fq(C)2300 1304 y(N)2376 1269 y Fo(\017)2408 1304 y Fm(g)f(\032)h Fq(y)2601 1316 y Fo(i)2647 1304 y Fm(\001)18 b Fq(A)2750 1269 y Fk(0)2774 1304 y Fq(:)250 1548 y Fs(Let)27 b Fq(~)-41 b(u)24 b Fs(=)h(\()p Fq(y)634 1560 y Fn(1)671 1548 y Fq(;)14 b(y)749 1560 y Fn(2)786 1548 y Fq(;)g(:)g(:)g(:)f(;)h(y)1011 1560 y Fo(d)1050 1548 y Fs(\))1082 1518 y Fo(T)1134 1548 y Fs(.)40 b(As)28 b(b)r(efore,)h(w)n(e)f(de\014ne)h(a)f(matrix)g Fq(T)2360 1518 y Fk(0)2407 1548 y Fm(2)d Fq(M)2568 1560 y Fo(d)2606 1548 y Fs(\()p Fp(F)2683 1560 y Fo(p)2727 1548 y Fs(\))k(where)f(the)h(en)n(try)f(in)h(the)g Fq(j)5 b Fs(th)150 1647 y(ro)n(w,)26 b Fq(k)s Fs(th)i(column)g(is)1719 1772 y Fq(T)1780 1738 y Fk(0)1768 1793 y Fo(j;k)1906 1772 y Fs(=)50 b Fq(a)2065 1729 y Fn(\()p Fo(k)q Fn(\))2065 1795 y Fo(j)s(i)2158 1772 y Fq(:)150 1908 y Fs(Again)27 b(\()424 1908 y SDict begin H.S end 424 1908 a 0 TeXcolorgray Fs(3)p 0 TeXcolorgray 466 1855 a SDict begin H.R end 466 1855 a 466 1908 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 466 1908 a Fs(\))h(b)r(ecomes)150 2056 y(\(6\))1734 2056 y SDict begin H.S end 1734 2056 a 1734 2056 a SDict begin 12 H.A end 1734 2056 a 1734 2056 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.6) cvn /DEST pdfmark end 1734 2056 a Fq(T)1795 2022 y Fk(0)1814 2056 y Fq(~)-38 b(v)54 b Fs(=)49 b Fq(~)-41 b(u)o(y)2118 2022 y Fk(0)2115 2077 y Fo(i)2143 2056 y Fq(;)150 2204 y Fs(and)32 b(from)f(\()548 2204 y SDict begin H.S end 548 2204 a 0 TeXcolorgray Fs(4)p 0 TeXcolorgray 590 2151 a SDict begin H.R end 590 2151 a 590 2204 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 590 2204 a Fs(\),)i(all)f(the)g(en)n(tries)f(of)h(adj\()p Fq(T)1520 2174 y Fk(0)1543 2204 y Fs(\))g(are)f(b)r(ounded)h(b)n(y)g Fq(C)6 b Fs(.)50 b(Using)32 b(the)g(same)f(argumen)n(t)g(that)h(w)n(e)g (did)150 2304 y(earlier)23 b(for)g(the)i(matrix)e Fq(T)12 b Fs(,)24 b(w)n(e)g(can)g(deduce)g(that)h Fq(T)1817 2274 y Fk(0)1863 2304 y Fs(is)f(also)f(in)n(v)n(ertible)h(since)g Fq(A)g Fs(is)g(prop)r(er.)35 b(Multiplying)25 b(b)r(oth)150 2403 y(sides)i(of)h(\()476 2403 y SDict begin H.S end 476 2403 a 0 TeXcolorgray Fs(6)p 0 TeXcolorgray 518 2350 a SDict begin H.R end 518 2350 a 518 2403 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 0]BorderArrayPatch/Subtype /Link/Dest (equation.2.6) cvn H.B /ANN pdfmark end 518 2403 a Fs(\))g(on)f(the)h(left)g(b)n(y)f(adj\()p Fq(T)1302 2373 y Fk(0)1325 2403 y Fs(\))h(w)n(e)f(get)1061 2573 y Fq(A)1123 2585 y Fn(2)1183 2573 y Fs(:=)c Fm(f)p Fs(det\()p Fq(T)1544 2539 y Fk(0)1566 2573 y Fs(\))1612 2494 y Fl(X)1657 2671 y Fo(j)1746 2573 y Fq(b)1782 2585 y Fo(j)1817 2573 y Fq(x)1864 2585 y Fo(j)1922 2573 y Fs(:)g Fm(j)p Fq(b)2027 2585 y Fo(j)2062 2573 y Fm(j)g Fq(<)2220 2517 y(c)p 2206 2554 66 4 v 2206 2630 a(C)2281 2573 y(N)2357 2539 y Fn(1)p Fk(\000)p Fo(\017)2474 2573 y Fm(g)f(\032)h Fq(A)c Fm(\001)f Fq(y)2792 2539 y Fk(0)2789 2593 y Fo(i)2816 2573 y Fq(:)150 2804 y Fs(Hence)1392 2917 y Fq(A)1454 2929 y Fn(1)1491 2917 y Fq(A)1553 2929 y Fn(2)1614 2917 y Fm(\032)23 b Fq(y)1743 2929 y Fo(i)1788 2917 y Fm(\001)c Fs(\()p Fq(A)1924 2883 y Fk(0)1948 2917 y Fq(A)p Fs(\))g Fm(\001)f Fq(y)2146 2883 y Fk(0)2143 2938 y Fo(i)2194 2917 y Fm(\032)k Fq(y)2322 2929 y Fo(i)2350 2917 y Fq(B)t(y)2461 2883 y Fk(0)2458 2938 y Fo(i)2485 2917 y Fq(:)150 3048 y Fs(No)n(w)27 b(for)g(an)n(y)g Fq(j;)14 b(k)26 b Fm(2)e Fs([)p Fq(d)p Fs(])p Fq(;)k Fs(w)n(e)f(ha)n(v)n(e)1221 3278 y(det\()p Fq(T)12 b Fs(\)det\()p Fq(T)1669 3243 y Fk(0)1692 3278 y Fs(\))p Fq(x)1771 3290 y Fo(j)1806 3278 y Fq(x)1853 3290 y Fo(k)1945 3278 y Fs(=)51 b Fq(y)2102 3290 y Fo(i)2143 3136 y Fl( )2251 3174 y Fo(d)2208 3199 y Fl(X)2216 3377 y Fo(l)p Fn(=1)2342 3278 y Fq(s)2381 3234 y Fn(\()p Fo(l)p Fn(\))2381 3303 y Fo(j)s(k)2459 3278 y Fq(x)2506 3290 y Fo(l)2531 3136 y Fl(!)2611 3278 y Fq(y)2655 3243 y Fk(0)2652 3298 y Fo(i)150 3500 y Fs(and)27 b(the)h(set)1298 3640 y Fm(f)p Fs(det\()p Fq(T)12 b Fs(\)det\()p Fq(T)1788 3606 y Fk(0)1810 3640 y Fs(\))p Fq(w)r(x)1950 3652 y Fo(j)1987 3640 y Fq(x)2034 3652 y Fo(k)2098 3640 y Fs(:)23 b Fm(j)p Fq(w)r Fm(j)h Fq(<)2376 3584 y(cc)2448 3554 y Fk(0)p 2372 3621 103 4 v 2372 3697 a Fq(C)2437 3673 y Fn(2)2485 3640 y Fq(N)9 b Fm(g)150 3799 y Fs(will)24 b(b)r(e)g(a)f(subset)h(of)f Fq(y)859 3811 y Fo(i)887 3799 y Fs(\(2)p Fq(B)t Fs(\))p Fq(y)1104 3769 y Fk(0)1101 3821 y Fo(i)1152 3799 y Fs(from)g(the)h (same)f(argumen)n(t)g(as)g(the)h(one)f(at)h(the)g(b)r(eginning)f(of)h (our)f(pro)r(of.)35 b(Again)150 3899 y(from)27 b(the)h(fact)g(that)g Fq(B)k Fs(is)27 b(6-isolated)f(w)n(e)h(w)n(ould)h(ha)n(v)n(e)1507 4047 y(max)p Fm(fj)p Fq(s)1766 4004 y Fn(\()p Fo(l)p Fn(\))1766 4072 y Fo(j)s(k)1843 4047 y Fm(j)23 b Fs(:)g Fq(l)h Fm(2)g Fs([)p Fq(d)p Fs(])p Fm(g)f Fq(<)f(C)2369 4012 y Fk(0)150 4202 y Fs(for)27 b(some)g Fq(C)550 4172 y Fk(0)597 4202 y Fs(=)22 b Fq(C)749 4172 y Fk(0)773 4202 y Fs(\()p Fq(c;)14 b(c)914 4172 y Fk(0)937 4202 y Fq(;)g(d)p Fs(\).)38 b(Th)n(us)27 b(there)g(exists)h Fq(T)1811 4214 y Fo(j)1868 4202 y Fm(2)c Fq(M)2028 4214 y Fo(d)2066 4202 y Fs(\()p Fp(F)2143 4214 y Fo(p)2187 4202 y Fs(\))k(whose)f(en)n(tries)g(are)g(b)r(ounded)h(b)n(y)f Fq(C)3416 4172 y Fk(0)3467 4202 y Fs(so)g(that)1391 4350 y(det\()p Fq(T)12 b Fs(\)det\()p Fq(T)1839 4316 y Fk(0)1862 4350 y Fs(\))p Fq(x)1941 4362 y Fo(j)1973 4350 y Fq(~)-39 b(v)54 b Fs(=)23 b Fq(y)2199 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