2 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
3 %%Title: diag_rational.dvi
6 %%BoundingBox: 0 0 596 842
7 %%DocumentPaperSizes: a4
9 %DVIPSCommandLine: dvips -Pljfour -o diag_rational.ps diag_rational.dvi
10 %DVIPSParameters: dpi=600, compressed, comments removed
11 %DVIPSSource: TeX output 1997.02.16:1419
12 %%BeginProcSet: texc.pro
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15 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
16 ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
17 isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
18 hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
19 TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
20 forall round exch round exch]setmatrix}N /@landscape{/isls true N}B
21 /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B
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27 N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup
28 length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{
29 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub
30 get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data
31 dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N
32 /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup
33 /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx
34 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff
35 setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff
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37 /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id
38 gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp
39 add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add
40 /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{
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42 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2
43 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string
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51 cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin
52 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul
53 add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict
54 /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook
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56 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for
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58 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V
59 {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7
60 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false}
61 ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
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63 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform
64 round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg
65 rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail
66 {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M}
67 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{
68 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{
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70 a}B /bos{/SS save N}B /eos{SS restore}B end
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941 B7FCA4C601E0C7FCB3A3ED01E0A9150302F013C0137F150790393FF80F8090391FFC1F00
942 6DB5FC6D13FC01015B9038003FE023407EBE2C>116 D<B6903803FFFCA5000101E09038
943 003E006C163C80017F5D8017F8013F5D6E1301011F5D6E1303010F5D6E13076D5DED800F
944 6D92C7FC15C05E6DEBE01E163E6D143CEDF07C027F1378EDF8F8023F5B15FD021F5B15FF
945 6E5BA36E5BA26E90C8FCA26E5AA26E5AA21578362C7EAB3B>118
949 %%Feature: *Resolution 600dpi
958 1 0 bop 257 573 a Fl(Theorem:)404 801 y Fk(Let)32 b Fj(M)44
959 b Fk(b)s(e)33 b(a)f(torsion-free)f Fj(R)q Fk(-mo)s(dule,)g(and)i
960 Fj(d)27 b(>)h Fk(0.)43 b(Let)920 1021 y Fj(f)38 b Fk(=)1177
961 938 y Fi(X)1110 1112 y Fh(n)1153 1121 y Fg(1)1187 1112
962 y Fh(;:::;n)1330 1124 y Ff(d)1381 1021 y Fj(a)1432 1036
963 y Fh(n)1475 1045 y Fg(1)1509 1036 y Fh(;:::;n)1652 1048
964 y Ff(d)1707 1021 y Fj(x)1762 980 y Fh(n)1805 989 y Fg(1)1762
965 1046 y Fe(1)1861 1021 y Fd(\001)17 b(\001)g(\001)e Fj(x)2049
966 975 y Fh(n)2092 987 y Ff(d)2049 1046 y Fh(d)2160 1021
967 y Fd(2)28 b Fj(M)10 b Fk([[)p Fj(x)2467 1036 y Fe(1)2508
968 1021 y Fj(;)17 b(:)g(:)g(:)f(x)2738 1036 y Fh(d)2779
969 1021 y Fk(]])257 1316 y(b)s(e)26 b(a)g(rational)d(function,)k(i.e.)41
970 b(there)27 b(are)e Fj(P)42 b Fd(2)28 b Fj(M)10 b Fk([)p
971 Fj(x)2167 1331 y Fe(1)2207 1316 y Fj(;)17 b(:)g(:)g(:)f(;)h(x)2481
972 1331 y Fh(d)2522 1316 y Fk(])26 b(and)g Fj(Q)i Fd(2)g
973 Fj(R)q Fk([)p Fj(x)3114 1331 y Fe(1)3154 1316 y Fj(;)17
974 b(:)g(:)g(:)f(;)h(x)3428 1331 y Fh(d)3468 1316 y Fk(])257
975 1437 y(with)33 b Fj(Q)p Fk(\(0)p Fj(;)17 b(:)g(:)g(:)e(;)i
976 Fk(0\))27 b(=)h(1)k(and)h Fj(Q)22 b Fd(\001)g Fj(f)39
977 b Fk(=)27 b Fj(P)14 b Fk(.)43 b(Then)34 b(the)f(full)d(diagonal)h(of)h
978 Fj(f)1532 1714 y(g)f Fk(=)1744 1606 y Fc(1)1720 1631
979 y Fi(X)1713 1813 y Fh(n)p Fe(=0)1863 1714 y Fj(a)1914
980 1729 y Fh(n;:::)n(;n)2118 1714 y Fj(x)2173 1673 y Fh(n)2173
981 1739 y Fe(1)257 2004 y Fk(is)h(a)h(D-\014nite)e(elemen)m(t)h(of)h
982 Fj(M)10 b Fk([[)p Fj(x)1478 2019 y Fe(1)1518 2004 y Fk(]],)33
983 b(w.r.t.)44 b Fj(R)q Fk([)p Fj(x)2060 2019 y Fe(1)2100
984 2004 y Fk(])33 b(and)g Fd(f)p Fj(@)2451 2019 y Fh(x)2491
985 2028 y Fg(1)2529 2004 y Fd(g)p Fk(.)257 2232 y Fl(Pro)s(of:)404
986 2460 y Fk(F)-8 b(rom)30 b(the)j(h)m(yp)s(otheses,)i Fj(M)10
987 b Fk([[)p Fj(x)1564 2475 y Fe(1)1605 2460 y Fj(;)17 b(:)g(:)g(:)e(;)i
988 (x)1878 2475 y Fh(d)1919 2460 y Fk(]])32 b(is)g(a)g(torsion-free)f
989 (di\013eren)m(tial)g(mo)s(dule)257 2581 y(o)m(v)m(er)43
990 b Fj(R)q Fk([)p Fj(x)632 2596 y Fe(1)672 2581 y Fj(;)17
991 b(:)g(:)g(:)f(;)h(x)946 2596 y Fh(d)986 2581 y Fk(])42
992 b(w.r.t.)71 b(the)42 b(deriv)-5 b(ativ)m(es)42 b Fd(f)p
993 Fj(@)2126 2596 y Fh(x)2166 2605 y Fg(1)2204 2581 y Fj(;)17
994 b(:)g(:)g(:)f(;)h(@)2474 2596 y Fh(x)2514 2608 y Ff(d)2554
995 2581 y Fd(g)p Fk(,)44 b(and)e Fj(f)52 b Fk(is)41 b(a)g(D-\014nite)257
996 2701 y(elemen)m(t)26 b(of)f Fj(M)10 b Fk([[)p Fj(x)928
997 2716 y Fe(1)969 2701 y Fj(;)17 b(:)g(:)g(:)e(;)i(x)1242
998 2716 y Fh(d)1283 2701 y Fk(]])26 b(o)m(v)m(er)g Fj(R)q
999 Fk([)p Fj(x)1721 2716 y Fe(1)1761 2701 y Fj(;)17 b(:)g(:)g(:)f(;)h(x)
1000 2035 2716 y Fh(d)2076 2701 y Fk(])25 b(w.r.t.)42 b Fd(f)p
1001 Fj(@)2498 2716 y Fh(x)2538 2725 y Fg(1)2576 2701 y Fj(;)17
1002 b(:)g(:)g(:)f(;)h(@)2846 2716 y Fh(x)2886 2728 y Ff(d)2927
1003 2701 y Fd(g)p Fk(.)40 b(No)m(w)26 b(apply)257 2821 y(the)33
1004 b(general)f(diagonal)f(theorem)h(\([1],)g(section)h(2.18\))f
1005 Fj(d)22 b Fd(\000)g Fk(1)33 b(times.)257 3050 y Fl(Theorem:)404
1006 3278 y Fk(Let)g Fj(R)i Fk(b)s(e)e(an)g(in)m(tegral)f(domain)g(of)h(c)m
1007 (haracteristic)g(0)g(and)g Fj(M)44 b Fk(sim)m(ultaneously)32
1008 b(a)257 3398 y(torsion-free)38 b Fj(R)q Fk(-mo)s(dule)g(and)h(a)f(comm)
1009 m(utativ)m(e)g Fj(R)q Fk(-algebra)g(without)g(zero)i(divisors.)257
1010 3519 y(Let)1223 3639 y Fj(f)e Fk(=)1460 3556 y Fi(X)1413
1011 3739 y Fh(m;n)p Fc(\025)p Fe(0)1644 3639 y Fj(a)1695
1012 3654 y Fh(m;n)1825 3639 y Fj(x)1880 3598 y Fh(m)1947
1013 3639 y Fj(y)1999 3598 y Fh(n)2072 3639 y Fd(2)28 b Fj(M)10
1014 b Fk([[)p Fj(x;)17 b(y)t Fk(]])257 3891 y(b)s(e)33 b(a)f(rational)f
1015 (function.)43 b(Then)33 b(the)g(diagonal)d(of)j Fj(f)1571
1016 4164 y(g)e Fk(=)1784 4056 y Fc(1)1759 4081 y Fi(X)1752
1017 4262 y Fh(n)p Fe(=0)1902 4164 y Fj(a)1953 4179 y Fh(n;n)2079
1018 4164 y Fj(x)2134 4122 y Fh(n)257 4453 y Fk(is)h(algebraic)f(o)m(v)m(er)
1019 j Fj(R)q Fk([)p Fj(x)p Fk(].)257 4681 y Fl(Motiv)-6 b(ation)36
1020 b(of)i(pro)s(of:)404 4909 y Fk(The)30 b(usual)g(pro)s(of)f(\([2]\))g
1021 (uses)j(complex)d(analysis)g(and)h(w)m(orks)h(only)e(for)h
1022 Fj(R)e Fk(=)g Fj(M)38 b Fk(=)257 5030 y Fj(C)7 b Fk(.)44
1023 b(The)33 b(idea)f(is)g(to)h(compute)1256 5294 y Fj(g)t
1024 Fk(\()p Fj(x)1400 5252 y Fe(2)1439 5294 y Fk(\))28 b(=)1664
1025 5226 y(1)p 1618 5270 141 4 v 1618 5362 a(2)p Fj(\031)t(i)1786
1026 5176 y Fi(I)1832 5365 y Fc(j)p Fh(z)s Fc(j)p Fe(=1)2018
1027 5294 y Fj(f)11 b Fk(\()p Fj(xz)t(;)2273 5226 y(x)p 2273
1028 5270 56 4 v 2276 5362 a(z)2338 5294 y Fk(\))2386 5226
1029 y Fj(dz)p 2386 5270 101 4 v 2412 5362 a(z)1852 5637 y
1032 2 1 bop 257 573 a Fk(This)30 b(in)m(tegral,)e(whose)i(in)m(tegrand)f
1033 (is)g(a)g(rational)e(function)h(in)h Fj(x)g Fk(and)h
1034 Fj(z)t Fk(,)g(is)f(calculated)257 693 y(using)41 b(the)g(residue)g
1035 (theorem.)68 b(Since)41 b Fj(f)11 b Fk(\()p Fj(x;)17
1036 b(y)t Fk(\))39 b(is)h(con)m(tin)m(uous)i(at)e(\(0)p Fj(;)17
1037 b Fk(0\),)42 b(there)f(is)f(a)257 814 y Fj(\016)32 b(>)c
1038 Fk(0)k(suc)m(h)i(that)e Fj(f)11 b Fk(\()p Fj(x;)17 b(y)t
1039 Fk(\))26 b Fd(6)p Fk(=)i Fd(1)k Fk(for)g Fd(j)p Fj(x)p
1040 Fd(j)27 b Fj(<)h(\016)t Fk(,)k Fd(j)p Fj(y)t Fd(j)26
1041 b Fj(<)i(\016)t Fk(.)43 b(It)33 b(follo)m(ws)e(that)h(for)g(all)f
1042 Fj(")c(>)g Fk(0)257 934 y(and)g Fd(j)p Fj(x)p Fd(j)g
1043 Fj(<)h(\016)t(")d Fk(all)g(the)h(p)s(oles)g(of)g Fj(f)11
1044 b Fk(\()p Fj(xz)t(;)1693 895 y Fh(x)p 1693 911 40 4 v
1045 1695 968 a(z)1743 934 y Fk(\))26 b(are)g(con)m(tained)h(in)e
1046 Fd(f)p Fj(z)32 b Fk(:)c Fd(j)p Fj(z)t Fd(j)g Fj(<)f(")p
1047 Fd(g)9 b([)g(f)p Fj(z)33 b Fk(:)28 b Fd(j)p Fj(z)t Fd(j)f
1048 Fj(>)267 1015 y Fe(1)p 267 1031 36 4 v 268 1089 a Fh(")313
1049 1054 y Fd(g)p Fk(.)49 b(Th)m(us)37 b(the)e(p)s(oles)f(of)h
1050 Fj(f)11 b Fk(\()p Fj(xz)t(;)1477 1015 y Fh(x)p 1477 1031
1051 40 4 v 1479 1089 a(z)1527 1054 y Fk(\),)35 b(all)e(algebraic)g
1052 (functions)i(of)f Fj(x)h Fk({)g(let's)f(call)f(them)257
1053 1175 y Fj(\020)300 1190 y Fe(1)339 1175 y Fk(\()p Fj(x)p
1054 Fk(\))p Fj(;)17 b(:)g(:)g(:)g(\020)689 1190 y Fh(s)725
1055 1175 y Fk(\()p Fj(x)p Fk(\))36 b({,)g(can)h(b)s(e)f(divided)f(up)h(in)m
1056 (to)f(those)i(for)e(whic)m(h)h Fd(j)p Fj(\020)2779 1190
1057 y Fh(i)2807 1175 y Fk(\()p Fj(x)p Fk(\))p Fd(j)d Fk(=)g
1058 Fj(O)s Fk(\()p Fd(j)p Fj(x)p Fd(j)p Fk(\))i(as)257 1295
1059 y Fj(x)43 b Fd(!)e Fk(0)g(and)g(those)g(for)g(whic)m(h)1580
1060 1256 y Fe(1)p 1502 1272 192 4 v 1502 1329 a Fc(j)p Fh(\020)1553
1061 1339 y Ff(i)1579 1329 y Fe(\()p Fh(x)p Fe(\))p Fc(j)1745
1062 1295 y Fk(=)h Fj(O)s Fk(\()p Fd(j)p Fj(x)p Fd(j)p Fk(\))d(as)j
1063 Fj(x)g Fd(!)f Fk(0.)69 b(It)40 b(follo)m(ws)g(from)g(the)257
1064 1429 y(residue)33 b(theorem)g(that)f(for)g Fd(j)p Fj(x)p
1065 Fd(j)c Fj(<)f(\016)1193 1672 y(g)t Fk(\()p Fj(x)1337
1066 1631 y Fe(2)1376 1672 y Fk(\))g(=)1711 1589 y Fi(X)1545
1067 1777 y Fh(\020)5 b Fe(=0)p Fc(_)p Fh(\020)g Fe(=)p Fh(O)r
1068 Fe(\()p Fc(j)p Fh(x)p Fc(j)p Fe(\))2014 1672 y Fk(Res)2028
1069 1732 y Fh(z)s Fe(=)p Fh(\020)2201 1672 y Fj(f)11 b Fk(\()p
1070 Fj(xz)t(;)2457 1605 y(x)p 2457 1649 56 4 v 2460 1741
1071 a(z)2522 1672 y Fk(\))257 1996 y(This)32 b(is)e(algebraic)g(o)m(v)m(er)
1072 i Fj(C)7 b Fk(\()p Fj(x)p Fk(\).)43 b(Hence)32 b Fj(g)t
1073 Fk(\()p Fj(x)p Fk(\))f(is)f(algebraic)g(o)m(v)m(er)i
1074 Fj(C)7 b Fk(\()p Fj(x)2856 1960 y Fe(1)p Fh(=)p Fe(2)2966
1075 1996 y Fk(\),)32 b(hence)g(also)257 2117 y(algebraic)f(o)m(v)m(er)j
1076 Fj(C)7 b Fk(\()p Fj(x)p Fk(\).)257 2345 y Fl(Pro)s(of:)404
1077 2573 y Fk(Let)647 2826 y Fj(h)p Fk(\()p Fj(x;)17 b(z)t
1078 Fk(\))29 b(:=)e Fj(f)11 b Fk(\()p Fj(xz)t(;)1342 2759
1079 y(x)p 1342 2803 V 1345 2894 a(z)1407 2826 y Fk(\))28
1080 b(=)1649 2718 y Fc(1)1624 2743 y Fi(X)1576 2924 y Fh(m;n)p
1081 Fe(=0)1808 2826 y Fj(a)1859 2841 y Fh(m;n)1988 2826 y
1082 Fj(x)2043 2785 y Fh(m)p Fe(+)p Fh(n)2208 2826 y Fj(z)2257
1083 2785 y Fh(m)p Fc(\000)p Fh(n)2450 2826 y Fd(2)g Fj(M)10
1084 b Fk([[)p Fj(xz)t(;)17 b(xz)2954 2785 y Fc(\000)p Fe(1)3051
1085 2826 y Fk(]])257 3136 y(Then)48 b Fj(g)t Fk(\()p Fj(x)670
1086 3099 y Fe(2)709 3136 y Fk(\))e(is)g(the)h(co)s(e\016cien)m(t)g(of)f
1087 Fj(z)1730 3099 y Fe(0)1816 3136 y Fk(in)g Fj(h)p Fk(\()p
1088 Fj(x;)17 b(z)t Fk(\).)85 b(Let)47 b Fj(N)10 b Fk(\()p
1089 Fj(x;)17 b(z)t Fk(\))52 b(:=)f Fj(z)3092 3099 y Fh(d)3134
1090 3136 y Fj(Q)p Fk(\()p Fj(xz)t(;)3408 3096 y Fh(x)p 3408
1091 3112 40 4 v 3410 3170 a(z)3457 3136 y Fk(\))257 3256
1092 y(\(with)36 b Fj(d)e Fk(:=)h(max)o(\(deg)1111 3279 y
1093 Fh(y)1169 3256 y Fj(P)s(;)17 b Fk(deg)1426 3279 y Fh(y)1484
1094 3256 y Fj(Q)p Fk(\)\))37 b(b)s(e)f(\\the)h(denominator")e(of)h
1095 Fj(h)p Fk(\()p Fj(x;)17 b(z)t Fk(\).)56 b(W)-8 b(e)37
1096 b(ha)m(v)m(e)257 3376 y Fj(N)10 b Fk(\()p Fj(x;)17 b(z)t
1097 Fk(\))31 b Fd(2)e Fj(R)q Fk([)p Fj(x;)17 b(z)t Fk(])34
1098 b(and)g Fj(N)39 b Fd(6)p Fk(=)29 b(0)k(\(b)s(ecause)i
1099 Fj(N)10 b Fk(\(0)p Fj(;)17 b(z)t Fk(\))29 b(=)g Fj(z)2390
1100 3340 y Fh(d)2431 3376 y Fk(\).)46 b(Let)33 b Fj(K)41
1101 b Fk(b)s(e)33 b(the)h(quotien)m(t)257 3497 y(\014eld)f(of)f
1102 Fj(R)q Fk(.)43 b(Th)m(us)34 b Fj(N)10 b Fk(\()p Fj(x;)17
1103 b(z)t Fk(\))29 b Fd(2)g Fj(K)7 b Fk([)p Fj(x)p Fk(][)p
1104 Fj(z)t Fk(])23 b Fd(n)f(f)p Fk(0)p Fd(g)p Fk(.)404 3617
1105 y(It)34 b(is)g(w)m(ell-kno)m(wn)g(\(see)h([3],)g(p.64,)f(or)g([4],)h(c)
1106 m(hap.)49 b(IV,)34 b Fd(x)p Fk(2,)h(prop.)48 b(8,)35
1107 b(or)f([5],)g(c)m(hap.)257 3737 y(I)s(I)s(I,)k Fd(x)p
1108 Fk(1\))g(that)g(the)h(splitting)c(\014eld)j(of)g Fj(N)10
1109 b Fk(\()p Fj(x;)17 b(z)t Fk(\))39 b(o)m(v)m(er)g Fj(K)7
1110 b Fk(\()p Fj(x)p Fk(\))38 b(can)h(b)s(e)f(em)m(b)s(edded)h(in)m(to)257
1111 3858 y(a)46 b(\014eld)h Fj(L)p Fk(\(\()p Fj(x)775 3822
1112 y Fe(1)p Fh(=r)884 3858 y Fk(\)\),)j(where)d Fj(r)i Fk(is)d(a)g(p)s
1113 (ositiv)m(e)g(in)m(teger)g(and)h Fj(L)g Fk(is)e(a)i(\014nite-algebraic)
1114 257 3978 y(extension)39 b(\014eld)e(of)g Fj(K)7 b Fk(,)38
1115 b(i.e.)58 b(a)37 b(simple)f(algebraic)f(extension)k Fj(L)d
1116 Fk(=)f Fj(K)7 b Fk(\()p Fj(\013)q Fk(\))36 b(=)g Fj(K)7
1117 b(\013)3355 3942 y Fe(0)3419 3978 y Fk(+)257 4099 y Fd(\001)17
1118 b(\001)g(\001)k Fk(+)h Fj(K)7 b(\013)647 4062 y Fh(u)p
1119 Fc(\000)p Fe(1)782 4099 y Fk(.)423 4187 y Fi(f)404 4219
1120 y Fj(M)38 b Fk(:=)28 b(\()p Fj(R)22 b Fd(n)f(f)p Fk(0)p
1121 Fd(g)p Fk(\))1059 4183 y Fc(\000)p Fe(1)1173 4219 y Fd(\001)g
1122 Fj(M)43 b Fk(is)31 b(a)h Fj(K)7 b Fk(-v)m(ector)33 b(space)g(and)f(a)g
1123 (comm)m(utativ)m(e)f Fj(K)7 b Fk(-algebra)257 4339 y(without)32
1124 b(zero)g(divisors.)1238 4307 y Fi(c)1219 4339 y Fj(M)38
1125 b Fk(:=)1501 4307 y Fi(f)1482 4339 y Fj(M)11 b(\013)1650
1126 4303 y Fe(0)1710 4339 y Fk(+)20 b Fd(\001)d(\001)g(\001)i
1127 Fk(+)2059 4307 y Fi(f)2040 4339 y Fj(M)11 b(\013)2208
1128 4303 y Fh(u)p Fc(\000)p Fe(1)2374 4339 y Fk(is)32 b(an)g
1129 Fj(L)p Fk(-v)m(ector)g(space)h(and)f(a)257 4460 y(comm)m(utativ)m(e)g
1130 Fj(L)p Fk(-algebra)f(without)h(zero)h(divisors.)585 4768
1131 y Fi(c)566 4800 y Fj(M)11 b Fd(h)-17 b(h)p Fj(x;)17 b(z)t
1132 Fd(i)-17 b(i)83 b Fk(:=)1229 4768 y Fi(c)1210 4800 y
1133 Fj(M)11 b Fk([[)p Fj(x)1424 4759 y Fe(1)p Fh(=r)1556
1134 4800 y Fd(\001)21 b Fj(z)t(;)c(x)1753 4759 y Fe(1)p Fh(=r)1885
1135 4800 y Fd(\001)22 b Fj(z)1984 4759 y Fc(\000)p Fe(1)2079
1136 4800 y Fj(;)17 b(x)2178 4759 y Fe(1)p Fh(=r)2286 4800
1137 y Fk(]][)p Fj(x)2422 4759 y Fc(\000)p Fe(1)p Fh(=r)2587
1138 4800 y Fk(])1038 5008 y(=)1210 4862 y Fi(\()1280 4925
1139 y(X)1277 5099 y Fh(m;n)1419 5008 y Fj(c)1461 5023 y Fh(m;n)1590
1140 5008 y Fj(x)1645 4967 y Fh(m=r)1781 5008 y Fj(z)1830
1141 4967 y Fh(n)1905 5008 y Fk(:)28 b Fj(c)2002 5023 y Fh(m;n)2159
1142 5008 y Fd(6)p Fk(=)f(0)h Fd(\))f(j)p Fj(n)p Fd(j)g(\024)h
1143 Fj(m)23 b Fk(+)f Fj(O)s Fk(\(1\))3121 4862 y Fi(\))257
1144 5323 y Fk(is)32 b(an)h Fj(L)p Fk(-algebra)e(whic)m(h)i(con)m(tains)1612
1145 5290 y Fi(c)1593 5323 y Fj(M)11 b Fk(\(\()p Fj(x)1829
1146 5287 y Fe(1)p Fh(=r)1938 5323 y Fk(\)\).)1852 5637 y(2)p
1149 3 2 bop 404 573 a Fk(Since)24 b Fj(N)10 b Fk(\()p Fj(x;)17
1150 b(z)t Fk(\))25 b(splits)d(in)m(to)h(linear)g(factors)g(in)g
1151 Fj(L)p Fk(\(\()p Fj(x)2295 537 y Fe(1)p Fh(=r)2404 573
1152 y Fk(\)\)[)p Fj(z)t Fk(],)k Fj(N)10 b Fk(\()p Fj(x;)17
1153 b(z)t Fk(\))29 b(=)e Fj(l)3129 506 y Fi(Q)3208 533 y
1154 Fh(s)3208 597 y(i)p Fe(=1)3326 573 y Fk(\()p Fj(z)9 b
1155 Fd(\000)257 712 y Fj(\020)300 727 y Fh(i)328 712 y Fk(\()p
1156 Fj(x)p Fk(\)\))497 676 y Fh(k)534 686 y Ff(i)565 712
1157 y Fk(,)37 b(there)f(exists)h(a)f(partial)e(fraction)h(decomp)s(osition)
1158 f(of)i Fj(h)p Fk(\()p Fj(x;)17 b(z)t Fk(\))34 b(=)3116
1159 656 y Fh(P)10 b Fe(\()p Fh(xz)s(;)3304 629 y Ff(x)p 3303
1160 641 35 4 v 3305 682 a(z)3348 656 y Fe(\))p 3116 689 261
1161 4 v 3116 746 a Fh(Q)p Fe(\()p Fh(xz)s(;)3305 719 y Ff(x)p
1162 3304 731 35 4 v 3306 772 a(z)3348 746 y Fe(\))3419 712
1163 y Fk(=)267 839 y Fh(z)303 815 y Ff(d)339 839 y Fh(P)g
1164 Fe(\()p Fh(xz)s(;)527 811 y Ff(x)p 526 823 V 528 865
1165 a(z)571 839 y Fe(\))p 267 871 331 4 v 326 929 a Fh(N)d
1166 Fe(\()p Fh(x;z)s Fe(\))641 894 y Fk(in)773 862 y Fi(c)754
1167 894 y Fj(M)k Fd(h)-17 b(h)p Fj(x;)17 b(z)t Fd(i)-17 b(i)p
1168 Fk(:)985 1208 y Fj(h)p Fk(\()p Fj(x;)17 b(z)t Fk(\))29
1169 b(=)1448 1100 y Fh(l)1399 1125 y Fi(X)1397 1308 y Fh(j)t
1170 Fe(=0)1536 1208 y Fj(P)1599 1223 y Fh(j)1636 1208 y Fk(\()p
1171 Fj(x)p Fk(\))p Fj(z)1816 1167 y Fh(j)1876 1208 y Fk(+)2017
1172 1100 y Fh(s)1974 1125 y Fi(X)1976 1308 y Fh(i)p Fe(=1)2143
1173 1098 y Fh(k)2180 1108 y Ff(i)2114 1125 y Fi(X)2110 1310
1174 y Fh(k)r Fe(=1)2371 1141 y Fj(P)2434 1156 y Fh(i;k)2520
1175 1141 y Fk(\()p Fj(x)p Fk(\))p 2265 1185 493 4 v 2265
1176 1277 a(\()p Fj(z)e Fd(\000)c Fj(\020)2518 1292 y Fh(i)2546
1177 1277 y Fk(\()p Fj(x)p Fk(\)\))2715 1248 y Fh(k)257 1521
1178 y Fk(with)33 b Fj(P)543 1536 y Fh(j)579 1521 y Fk(\()p
1179 Fj(x)p Fk(\))p Fj(;)17 b(P)817 1536 y Fh(i;k)903 1521
1180 y Fk(\()p Fj(x)p Fk(\))28 b Fd(2)1175 1489 y Fi(c)1156
1181 1521 y Fj(M)11 b Fk(\(\()p Fj(x)1392 1485 y Fe(1)p Fh(=r)1501
1182 1521 y Fk(\)\).)404 1641 y(Recall)26 b(that)h(w)m(e)h(are)g(lo)s(oking)
1183 d(for)i(the)h(co)s(e\016cien)m(t)g(of)f Fj(z)2443 1605
1184 y Fe(0)2511 1641 y Fk(in)g Fj(h)p Fk(\()p Fj(x;)17 b(z)t
1185 Fk(\).)42 b(W)-8 b(e)28 b(compute)257 1762 y(it)k(separately)h(for)f
1186 (eac)m(h)h(summand.)404 1882 y(If)f Fj(\020)544 1897
1187 y Fh(i)572 1882 y Fk(\()p Fj(x)p Fk(\))c(=)g Fj(ax)941
1188 1846 y Fh(m=r)1099 1882 y Fk(+)22 b Fj(:::)33 b Fk(with)f
1189 Fj(a)c Fd(2)g Fj(L)22 b Fd(n)g(f)p Fk(0)p Fd(g)p Fk(,)32
1190 b Fj(m)c(>)g Fk(0,)k(or)h Fj(\020)2562 1897 y Fh(i)2589
1191 1882 y Fk(\()p Fj(x)p Fk(\))28 b(=)g(0,)k(w)m(e)i(ha)m(v)m(e)1097
1192 2165 y(1)p 875 2209 V 875 2300 a(\()p Fj(z)27 b Fd(\000)22
1193 b Fj(\020)1127 2315 y Fh(i)1155 2300 y Fk(\()p Fj(x)p
1194 Fk(\)\))1324 2272 y Fh(k)1460 2232 y Fk(=)1651 2165 y(1)p
1195 1629 2209 93 4 v 1629 2300 a Fj(z)1678 2272 y Fh(k)1753
1196 2232 y Fd(\001)2031 2165 y Fk(1)p 1813 2209 485 4 v 1813
1197 2242 a Fi(\020)1863 2338 y Fk(1)g Fd(\000)2043 2292 y
1198 Fh(\020)2074 2302 y Ff(i)2101 2292 y Fe(\()p Fh(x)p Fe(\))p
1199 2043 2315 152 4 v 2101 2373 a Fh(z)2205 2242 y Fi(\021)2255
1200 2265 y Fh(k)1460 2588 y Fk(=)1651 2521 y(1)p 1629 2565
1201 93 4 v 1629 2657 a Fj(z)1678 2628 y Fh(k)1753 2588 y
1202 Fd(\001)1829 2480 y Fc(1)1804 2505 y Fi(X)1803 2688 y
1203 Fh(j)t Fe(=0)1942 2442 y Fi( )2008 2521 y Fj(k)j Fd(\000)e
1204 Fk(1)f(+)g Fj(j)2091 2657 y(k)j Fd(\000)e Fk(1)2399 2442
1205 y Fi(!)16 b( )2557 2521 y Fj(\020)2600 2536 y Fh(i)2628
1206 2521 y Fk(\()p Fj(x)p Fk(\))p 2557 2565 203 4 v 2633
1207 2657 a Fj(z)2769 2442 y Fi(!)2835 2465 y Fh(j)1460 2891
1208 y Fk(=)1645 2783 y Fc(1)1620 2808 y Fi(X)1619 2991 y
1209 Fh(j)t Fe(=0)1758 2745 y Fi( )1824 2824 y Fj(k)25 b Fd(\000)e
1210 Fk(1)f(+)g Fj(j)1907 2960 y(k)j Fd(\000)e Fk(1)2214 2745
1211 y Fi(!)2290 2824 y Fj(\020)2333 2839 y Fh(i)2361 2824
1212 y Fk(\()p Fj(x)p Fk(\))2492 2788 y Fh(j)p 2290 2868 239
1213 4 v 2320 2960 a Fj(z)2369 2931 y Fh(k)r Fe(+)p Fh(j)257
1214 3211 y Fk(hence)34 b(the)f(co)s(e\016cien)m(t)h(of)e
1215 Fj(z)1312 3175 y Fe(0)1384 3211 y Fk(in)1567 3159 y Fh(P)1612
1216 3171 y Ff(i;k)1691 3159 y Fe(\()p Fh(x)p Fe(\))p 1508
1217 3187 336 4 v 1508 3245 a(\()p Fh(z)s Fc(\000)p Fh(\020)1657
1218 3255 y Ff(i)1683 3245 y Fe(\()p Fh(x)p Fe(\)\))1804 3226
1219 y Ff(k)1886 3211 y Fk(is)g(0.)404 3356 y(If)g Fj(\020)544
1220 3371 y Fh(i)572 3356 y Fk(\()p Fj(x)p Fk(\))c(=)g Fj(ax)941
1221 3320 y Fh(m=r)1099 3356 y Fk(+)22 b Fj(:::)33 b Fk(with)f
1222 Fj(a)c Fd(2)g Fj(L)22 b Fd(n)g(f)p Fk(0)p Fd(g)p Fk(,)32
1223 b Fj(m)c(<)g Fk(0,)k(w)m(e)i(ha)m(v)m(e)489 3578 y(1)p
1224 267 3622 493 4 v 267 3714 a(\()p Fj(z)27 b Fd(\000)c
1225 Fj(\020)520 3729 y Fh(i)548 3714 y Fk(\()p Fj(x)p Fk(\)\))717
1226 3685 y Fh(k)797 3646 y Fk(=)1085 3578 y(1)p 911 3622
1227 399 4 v 911 3714 a(\()p Fd(\000)p Fj(\020)1069 3729 y
1228 Fh(i)1097 3714 y Fk(\()p Fj(x)p Fk(\)\))1266 3685 y Fh(k)1319
1229 3646 y Fd(\001)1574 3578 y Fk(1)p 1357 3622 485 4 v 1357
1230 3656 a Fi(\020)1406 3752 y Fk(1)f Fd(\000)1645 3713 y
1231 Fh(z)p 1586 3729 152 4 v 1586 3786 a(\020)1617 3796 y
1232 Ff(i)1644 3786 y Fe(\()p Fh(x)p Fe(\))1748 3656 y Fi(\021)1798
1233 3678 y Fh(k)1878 3646 y Fk(=)2166 3578 y(1)p 1992 3622
1234 399 4 v 1992 3714 a(\()p Fd(\000)p Fj(\020)2150 3729
1235 y Fh(i)2178 3714 y Fk(\()p Fj(x)p Fk(\)\))2347 3685 y
1236 Fh(k)2400 3646 y Fd(\001)2453 3538 y Fc(1)2429 3563 y
1237 Fi(X)2428 3745 y Fh(j)t Fe(=0)2567 3500 y Fi( )2632 3578
1238 y Fj(k)k Fd(\000)c Fk(1)g(+)g Fj(j)2715 3714 y(k)k Fd(\000)c
1239 Fk(1)3023 3500 y Fi(!)16 b( )3257 3578 y Fj(z)p 3181
1240 3622 203 4 v 3181 3714 a(\020)3224 3729 y Fh(i)3252 3714
1241 y Fk(\()p Fj(x)p Fk(\))3393 3500 y Fi(!)3459 3522 y Fh(j)257
1242 4002 y Fk(hence)34 b(the)f(co)s(e\016cien)m(t)h(of)e
1243 Fj(z)1312 3965 y Fe(0)1384 4002 y Fk(in)1567 3950 y Fh(P)1612
1244 3962 y Ff(i;k)1691 3950 y Fe(\()p Fh(x)p Fe(\))p 1508
1245 3978 336 4 v 1508 4036 a(\()p Fh(z)s Fc(\000)p Fh(\020)1657
1246 4046 y Ff(i)1683 4036 y Fe(\()p Fh(x)p Fe(\)\))1804 4017
1247 y Ff(k)1886 4002 y Fk(is)2035 3950 y Fh(P)2080 3962 y
1248 Ff(i;k)2159 3950 y Fe(\()p Fh(x)p Fe(\))p 1994 3978 300
1249 4 v 1994 4036 a(\()p Fc(\000)p Fh(\020)2107 4046 y Ff(i)2134
1250 4036 y Fe(\()p Fh(x)p Fe(\)\))2255 4017 y Ff(k)2304 4002
1251 y Fk(.)404 4147 y(The)43 b(case)h Fj(\020)874 4162 y
1252 Fh(i)902 4147 y Fk(\()p Fj(x)p Fk(\))h(=)g Fj(ax)1305
1253 4111 y Fh(m=r)1471 4147 y Fk(+)29 b Fj(:::)43 b Fk(with)f
1254 Fj(a)k Fd(2)f Fj(L)30 b Fd(n)e(f)p Fk(0)p Fd(g)p Fk(,)45
1255 b Fj(m)h Fk(=)f(0,)g(cannot)e(o)s(ccur,)257 4267 y(b)s(ecause)34
1256 b(it)e(w)m(ould)g(imply)f(0)d(=)f Fj(N)10 b Fk(\(0)p
1257 Fj(;)17 b(\020)1716 4282 y Fh(i)1744 4267 y Fk(\(0\)\))27
1258 b(=)h Fj(N)10 b Fk(\(0)p Fj(;)17 b(a)p Fk(\))27 b(=)h
1259 Fj(a)2528 4231 y Fh(d)2569 4267 y Fj(:)404 4388 y Fk(Altogether)k(w)m
1260 (e)h(ha)m(v)m(e)514 4670 y Fj(g)t Fk(\()p Fj(x)658 4629
1261 y Fe(2)697 4670 y Fk(\))28 b(=)f([)p Fj(z)942 4629 y
1262 Fe(0)982 4670 y Fk(])p Fj(h)p Fk(\()p Fj(x;)17 b(z)t
1263 Fk(\))29 b(=)f Fj(P)1485 4685 y Fe(0)1524 4670 y Fk(\()p
1264 Fj(x)p Fk(\))22 b(+)1886 4587 y Fi(X)1839 4757 y Fg(1)p
1265 1785 4769 138 4 v 1785 4810 a Ff(\020)1813 4826 y(i)1840
1266 4810 y Fg(\()p Ff(x)p Fg(\))1933 4784 y Fe(=)p Fh(o)p
1267 Fe(\()p Fh(x)p Fe(\))2166 4559 y Fh(k)2203 4569 y Ff(i)2138
1268 4587 y Fi(X)2133 4771 y Fh(k)r Fe(=1)2347 4602 y Fj(P)2410
1269 4617 y Fh(i;k)2496 4602 y Fk(\()p Fj(x)p Fk(\))p 2288
1270 4647 399 4 v 2288 4738 a(\()p Fd(\000)p Fj(\020)2446
1271 4753 y Fh(i)2475 4738 y Fk(\()p Fj(x)p Fk(\)\))2644 4709
1272 y Fh(k)2724 4670 y Fd(2)2837 4638 y Fi(c)2818 4670 y
1273 Fj(M)11 b Fk(\(\()p Fj(x)3054 4629 y Fe(1)p Fh(=r)3163
1274 4670 y Fk(\)\))404 5027 y(Since)39 b(all)f Fj(\020)851
1275 5042 y Fh(i)878 5027 y Fk(\()p Fj(x)p Fk(\)\(in)h Fj(L)p
1276 Fk(\(\()p Fj(x)1365 4991 y Fe(1)p Fh(=r)1475 5027 y Fk(\)\)\))g(and)g
1277 (all)f Fj(P)2030 5042 y Fh(j)2066 5027 y Fk(\()p Fj(x)p
1278 Fk(\))p Fj(;)17 b(P)2304 5042 y Fh(i;k)2390 5027 y Fk(\()p
1279 Fj(x)p Fk(\))40 b(\(in)2739 4995 y Fi(c)2720 5027 y Fj(M)10
1280 b Fk(\(\()p Fj(x)2955 4991 y Fe(1)p Fh(=r)3064 5027 y
1281 Fk(\)\)\))39 b(are)h(al-)257 5147 y(gebraic)h(o)m(v)m(er)i
1282 Fj(K)7 b Fk(\()p Fj(x)p Fk(\),)44 b(the)e(same)f(holds)g(also)g(for)g
1283 Fj(g)t Fk(\()p Fj(x)2313 5111 y Fe(2)2352 5147 y Fk(\).)70
1284 b(Hence)43 b Fj(g)t Fk(\()p Fj(x)p Fk(\))e(is)g(algebraic)257
1285 5268 y(o)m(v)m(er)35 b Fj(K)7 b Fk(\()p Fj(x)650 5232
1286 y Fe(1)p Fh(=)p Fe(2)761 5268 y Fk(\),)33 b(hence)i(also)e(o)m(v)m(er)h
1287 Fj(K)7 b Fk(\()p Fj(x)p Fk(\).)47 b(After)34 b(clearing)e
1288 (denominators,)h(w)m(e)h(\014nally)257 5388 y(conclude)f(that)g
1289 Fj(g)t Fk(\()p Fj(x)p Fk(\))f(is)g(algebraic)f(o)m(v)m(er)i
1290 Fj(R)q Fk([)p Fj(x)p Fk(].)1852 5637 y(3)p eop
1292 4 3 bop 257 573 a Fb(References)62 792 y Fk(1)146 b(Bruno)46
1293 b(Haible:)69 b(D-\014nite)45 b(p)s(o)m(w)m(er)i(series)g(in)e(sev)m
1294 (eral)i(v)-5 b(ariables.)82 b Fa(Diploma)46 b(thesis,)501
1295 912 y(University)35 b(of)g(Karlsruhe,)f(June)h(1989.)44
1296 b Fk(Sections)33 b(2.18)f(and)h(2.20.)62 1116 y(2)146
1297 b(M.)28 b(L.)f(J.)h(Hautus,)h(D.)e(A.)g(Klarner:)40 b(The)28
1298 b(diagonal)d(of)i(a)g(double)g(p)s(o)m(w)m(er)h(series.)43
1299 b Fa(Duke)501 1236 y(Math.)i(J.)35 b Fl(38)j Fk(\(1971\),)31
1300 b(229-235.)62 1439 y(3)146 b(C.)37 b(Chev)-5 b(alley:)50
1301 b(In)m(tro)s(duction)35 b(to)h(the)g(theory)h(of)e(algebraic)f
1302 (functions)i(of)g(one)g(v)-5 b(ari-)501 1560 y(able.)43
1303 b Fa(Mathematic)-5 b(al)35 b(Surveys)f(VI.)h(A)n(meric)-5
1304 b(an)34 b(Mathematic)-5 b(al)34 b(So)-5 b(ciety.)62 1763
1305 y Fk(4)146 b(Jean-Pierre)33 b(Serre:)44 b(Corps)33 b(lo)s(caux.)43
1306 b Fa(Hermann.)h(Paris)35 b Fk(1968.)62 1967 y(5)146 b(Martin)38
1307 b(Eic)m(hler:)55 b(In)m(tro)s(duction)38 b(to)g(the)g(theory)h(of)f
1308 (algebraic)f(n)m(um)m(b)s(ers)i(and)f(func-)501 2087
1309 y(tions.)43 b Fa(A)-5 b(c)g(ademic)34 b(Pr)-5 b(ess.)45
1310 b(New)35 b(Y)-7 b(ork,)34 b(L)-5 b(ondon)34 b Fk(1966.)1852
1314 userdict /end-hook known{end-hook}if
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