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%!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: All.dvi %%Pages: 6 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips All.dvi -pp 65-70 -o bronstein.ps %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2000.11.09:1118 %%BeginProcSet: texc.pro %! /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 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1997 - Patch 2 %itransform translate pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a itransform pathbbox /y2 ED a Div ceiling cvi /x2 ED /y1 ED a % DG/SR modification end Div cvi /x1 ED /y2 y2 y1 sub def clip newpath 2 setlinecap systemdict /setstrokeadjust known { true setstrokeadjust } if x2 x1 sub 1 add { x1 % DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) % a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore } % def a mul y1 moveto 0 y2 rlineto stroke /x1 x1 1 add def } repeat grestore pop pop } def % DG/SR modification end /BeginArrow { ADict begin /@mtrx CM def gsave 2 copy T 2 index sub neg exch 3 index sub exch Atan rotate newpath } def /EndArrow { @mtrx setmatrix CP grestore end } def /Arrow { CLW mul add dup 2 div /w ED mul dup /h ED mul /a ED { 0 h T 1 -1 scale } if w neg h moveto 0 0 L w h L w neg a neg rlineto gsave fill grestore } def /Tbar { CLW mul add /z ED z -2 div CLW 2 div moveto z 0 rlineto stroke 0 CLW moveto } def 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ArrowA /n n 2 sub def n { Lineto } repeat CP 4 2 roll ArrowB L pop pop } if } def /Arcto { /a [ 6 -2 roll ] cvx def a r /arcto load stopped { 5 } { 4 } ifelse { pop } repeat a } def /CheckClosed { dup n 2 mul 1 sub index eq 2 index n 2 mul 1 add index eq and { pop pop /n n 1 sub def } if } def /Polygon { NArray n 2 eq { 0 0 /n 3 def } if n 3 lt { n { pop pop } repeat } { n 3 gt { CheckClosed } if n 2 mul -2 roll /y0 ED /x0 ED /y1 ED /x1 ED x1 y1 /x1 x0 x1 add 2 div def /y1 y0 y1 add 2 div def x1 y1 moveto /n n 2 sub def n { Lineto } repeat x1 y1 x0 y0 6 4 roll Lineto Lineto pop pop closepath } ifelse } def /Diamond { /mtrx CM def T rotate /h ED /w ED dup 0 eq { pop } { CLW mul neg /d ED /a w h Atan def /h d a sin Div h add def /w d a cos Div w add def } ifelse mark w 2 div h 2 div w 0 0 h neg w neg 0 0 h w 2 div h 2 div /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx setmatrix } def % DG modification begin - Jan. 15, 1997 %/Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup 0 eq { %pop } { CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 %div dup cos exch sin Div mul sub def } ifelse mark 0 d w neg d 0 h w d 0 %d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx %setmatrix } def /Triangle { /mtrx CM def translate rotate /h ED 2 div /w ED dup CLW mul /d ED /h h d w h Atan sin Div sub def /w w d h w Atan 2 div dup cos exch sin Div mul sub def mark 0 d w neg d 0 h w d 0 d /ArrowA { moveto } def /ArrowB { } def false Line closepath mtrx % DG/SR modification begin - Jun. 1, 1998 - Patch 3 (from Michael Vulis) % setmatrix } def setmatrix pop } def % DG/SR modification end /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth def } def /CCA { /y ED /x ED 2 copy y sub /dy1 ED x sub /dx1 ED /l1 dx1 dy1 Pyth def } def /CC { /l0 l1 def /x1 x dx sub def /y1 y dy sub def /dx0 dx1 def /dy0 dy1 def CCA /dx dx0 l1 c exp mul dx1 l0 c exp mul add def /dy dy0 l1 c exp mul dy1 l0 c exp mul add def /m dx0 dy0 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if a sub /y2 ED a add /y1 ED 2 copy gt { exch } if a sub /x2 ED a add /x1 ED 1 index 0 eq { pop pop Rect } { OvalFrame } ifelse } def /BezierNArray { /f ED counttomark 2 div dup cvi /n ED n eq not { exch pop } if n 1 sub neg 3 mod 3 add 3 mod { 0 0 /n n 1 add def } repeat f { ] aload /Points ED } { n 2 mul 1 add -1 roll pop } ifelse } def /OpenBezier { BezierNArray n 1 eq { pop pop } { ArrowA n 4 sub 3 idiv { 6 2 roll 4 2 roll curveto } repeat 6 2 roll 4 2 roll ArrowB curveto } ifelse } def /ClosedBezier { BezierNArray n 1 eq { pop pop } { moveto n 1 sub 3 idiv { 6 2 roll 4 2 roll curveto } repeat closepath } ifelse } def /BezierShowPoints { gsave Points aload length 2 div cvi /n ED moveto n 1 sub { lineto } repeat CLW 2 div SLW [ 4 4 ] 0 setdash stroke grestore } def /Parab { /y0 exch def /x0 exch def /y1 exch def /x1 exch def /dx x0 x1 sub 3 div def /dy y0 y1 sub 3 div def x0 dx sub y0 dy add x1 y1 ArrowA x0 dx add y0 dy add x0 2 mul x1 sub y1 ArrowB curveto /Points [ x1 y1 x0 y0 x0 2 mul x1 sub y1 ] def } def /Grid { newpath /a 4 string def /b ED /c ED /n ED cvi dup 1 lt { pop 1 } if /s ED s div dup 0 eq { pop 1 } if /dy ED s div dup 0 eq { pop 1 } if /dx ED dy div round dy mul /y0 ED dx div round dx mul /x0 ED dy div round cvi /y2 ED dx div round cvi /x2 ED dy div round cvi /y1 ED dx div round cvi /x1 ED /h y2 y1 sub 0 gt { 1 } { -1 } ifelse def /w x2 x1 sub 0 gt { 1 } { -1 } ifelse def b 0 gt { /z1 b 4 div CLW 2 div add def /Helvetica findfont b scalefont setfont /b b .95 mul CLW 2 div add def } if systemdict /setstrokeadjust known { true setstrokeadjust /t { } def } { /t { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } bind def } ifelse gsave n 0 gt { 1 setlinecap [ 0 dy n div ] dy n div 2 div setdash } { 2 setlinecap } ifelse /i x1 def /f y1 dy mul n 0 gt { dy n div 2 div h mul sub } if def /g y2 dy mul n 0 gt { dy n div 2 div h mul add } if def x2 x1 sub w mul 1 add dup 1000 gt { pop 1000 } if { i dx mul dup y0 moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1} {z1 z2 add neg} ifelse h 0 gt {b neg} {z1} ifelse rmoveto show grestore } if dup t f moveto g t L stroke /i i w add def } repeat grestore gsave n 0 gt % DG/SR modification begin - Nov. 7, 1997 - Patch 1 %{ 1 setlinecap [ 0 dx n div ] dy n div 2 div setdash } { 1 setlinecap [ 0 dx n div ] dx n div 2 div setdash } % DG/SR modification end { 2 setlinecap } ifelse /i y1 def /f x1 dx mul n 0 gt { dx n div 2 div w mul sub } if def /g x2 dx mul n 0 gt { dx n div 2 div w mul add } if def y2 y1 sub h mul 1 add dup 1000 gt { pop 1000 } if { newpath i dy mul dup x0 exch moveto b 0 gt { gsave c i a cvs dup stringwidth pop /z2 ED w 0 gt {z1 z2 add neg} {z1} ifelse h 0 gt {z1} {b neg} ifelse rmoveto show grestore } if dup f exch t moveto g exch t L stroke /i i h add def } repeat grestore } def /ArcArrow { /d ED /b ED /a ED gsave newpath 0 -1000 moveto clip newpath 0 1 0 0 b grestore c mul /e ED pop pop pop r a e d PtoC y add exch x add exch r 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mul abs y gt } ifelse { x x mul y y mul sub z z mul add sqrt z add } { q { x s div } { x c div } ifelse abs } ifelse a PtoC h1 add exch w1 add exch } def /BeginOL { dup (all) eq exch TheOL eq or { IfVisible not { Visible /IfVisible true def } if } { IfVisible { Invisible /IfVisible false def } if } ifelse } def /InitOL { /OLUnit [ 3000 3000 matrix defaultmatrix dtransform ] cvx def /Visible { CP OLUnit idtransform T moveto } def /Invisible { CP OLUnit neg exch neg exch idtransform T moveto } def /BOL { BeginOL } def /IfVisible true def } def end % END pstricks.pro %%EndProcSet %%BeginProcSet: pst-dots.pro %!PS-Adobe-2.0 %%Title: Dot Font for PSTricks 97 - Version 97, 93/05/07. %%Creator: Timothy Van Zandt tvz@princeton.edu %%Creation Date: May 7, 1993 10 dict dup begin /FontType 3 def /FontMatrix [ .001 0 0 .001 0 0 ] def /FontBBox [ 0 0 0 0 ] def /Encoding 256 array def 0 1 255 { Encoding exch /.notdef put } for Encoding dup (b) 0 get /Bullet put dup (c) 0 get /Circle put dup (C) 0 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CM def next end grestore } def /InitPnode { /Y ED /X ED /NodePos { NodeSep Cos mul NodeSep Sin mul } def } def /InitCnode { /r ED /Y ED /X ED /NodePos { NodeSep r add dup Cos mul exch Sin mul } def } def /GetRnodePos { Cos 0 gt { /dx r NodeSep add def } { /dx l NodeSep sub def } ifelse Sin 0 gt { /dy u NodeSep add def } { /dy d NodeSep sub def } ifelse dx Sin mul abs dy Cos mul abs gt { dy Cos mul Sin div dy } { dx dup Sin mul Cos Div } ifelse } def /InitRnode { /Y ED /X ED X sub /r ED /l X neg def Y add neg /d ED Y sub /u ED /NodePos { GetRnodePos } def } def /DiaNodePos { w h mul w Sin mul abs h Cos mul abs add Div NodeSep add dup Cos mul exch Sin mul } def /TriNodePos { Sin s lt { d NodeSep sub dup Cos mul Sin Div exch } { w h mul w Sin mul h Cos abs mul add Div NodeSep add dup Cos mul exch Sin mul } ifelse } def /InitTriNode { sub 2 div exch 2 div exch 2 copy T 2 copy 4 index index /d ED pop pop pop pop -90 mul rotate /NodeMtrx CM def /X 0 def /Y 0 def d sub abs neg /d ED d add /h ED 2 div h mul h d sub Div /w ED /s d w Atan sin def /NodePos { TriNodePos } def } def /OvalNodePos { /ww w NodeSep add def /hh h NodeSep add def Sin ww mul Cos hh mul Atan dup cos ww mul exch sin hh mul } def /GetCenter { begin X Y NodeMtrx transform CM itransform end } def /XYPos { dup sin exch cos Do /Cos ED /Sin ED /Dist ED Cos 0 gt { Dist Dist Sin mul Cos div } { Cos 0 lt { Dist neg Dist Sin mul Cos div neg } { 0 Dist Sin mul } ifelse } ifelse Do } def /GetEdge { dup 0 eq { pop begin 1 0 NodeMtrx dtransform CM idtransform exch atan sub dup sin /Sin ED cos /Cos ED /NodeSep ED NodePos NodeMtrx dtransform CM idtransform end } { 1 eq {{exch}} {{}} ifelse /Do ED pop XYPos } ifelse } def /AddOffset { 1 index 0 eq { pop pop } { 2 copy 5 2 roll cos mul add 4 1 roll sin mul sub exch } ifelse } def /GetEdgeA { NodeSepA AngleA NodeA NodeSepTypeA GetEdge OffsetA AngleA AddOffset yA add /yA1 ED xA add /xA1 ED } def /GetEdgeB { NodeSepB AngleB NodeB NodeSepTypeB GetEdge OffsetB AngleB AddOffset yB add /yB1 ED xB add /xB1 ED } def /GetArmA { ArmTypeA 0 eq { /xA2 ArmA AngleA cos mul xA1 add def /yA2 ArmA AngleA sin mul yA1 add def } { ArmTypeA 1 eq {{exch}} {{}} ifelse /Do ED ArmA AngleA XYPos OffsetA AngleA AddOffset yA add /yA2 ED xA add /xA2 ED } ifelse } def /GetArmB { ArmTypeB 0 eq { /xB2 ArmB AngleB cos mul xB1 add def /yB2 ArmB AngleB sin mul yB1 add def } { ArmTypeB 1 eq {{exch}} {{}} ifelse /Do ED ArmB AngleB XYPos OffsetB AngleB AddOffset yB add /yB2 ED xB add /xB2 ED } ifelse } def /InitNC { /b ED /a ED /NodeSepTypeB ED /NodeSepTypeA ED /NodeSepB ED /NodeSepA ED /OffsetB ED /OffsetA ED tx@NodeDict a known tx@NodeDict b known and dup { /NodeA a load def /NodeB b load def NodeA GetCenter /yA ED /xA ED NodeB GetCenter /yB ED /xB ED } if } def /LPutLine { 4 copy 3 -1 roll sub neg 3 1 roll sub Atan /NAngle ED 1 t sub mul 3 1 roll 1 t sub mul 4 1 roll t mul add /Y ED t mul add /X ED } def /LPutLines { mark LPutVar counttomark 2 div 1 sub /n ED t floor dup n gt { pop n 1 sub /t 1 def } { dup t sub neg /t ED } ifelse cvi 2 mul { pop } repeat LPutLine cleartomark } def /BezierMidpoint { /y3 ED /x3 ED /y2 ED /x2 ED /y1 ED /x1 ED /y0 ED /x0 ED /t ED /cx x1 x0 sub 3 mul def /cy y1 y0 sub 3 mul def /bx x2 x1 sub 3 mul cx sub def /by y2 y1 sub 3 mul cy sub def /ax x3 x0 sub cx sub bx sub def /ay y3 y0 sub cy sub by sub def ax t 3 exp mul bx t t mul mul add cx t mul add x0 add ay t 3 exp mul by t t mul mul add cy t mul add y0 add 3 ay t t mul mul mul 2 by t mul mul add cy add 3 ax t t mul mul mul 2 bx t mul mul add cx add atan /NAngle ED /Y ED /X ED } def /HPosBegin { yB yA ge { /t 1 t sub def } if /Y yB yA sub t mul yA add def } def /HPosEnd { /X Y yyA sub yyB yyA sub Div xxB xxA sub mul xxA add def /NAngle yyB yyA sub xxB xxA sub Atan def } def /HPutLine { HPosBegin /yyA ED /xxA ED /yyB ED /xxB ED HPosEnd } def /HPutLines { HPosBegin yB yA ge { /check { le } def } { /check { ge } def } ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { dup Y check { exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark HPosEnd } def /VPosBegin { xB xA lt { /t 1 t sub def } if /X xB xA sub t mul xA add def } def /VPosEnd { /Y X xxA sub xxB xxA sub Div yyB yyA sub mul yyA add def /NAngle yyB yyA sub xxB xxA sub Atan def } def /VPutLine { VPosBegin /yyA ED /xxA ED /yyB ED /xxB ED VPosEnd } def /VPutLines { VPosBegin xB xA ge { /check { le } def } { /check { ge } def } ifelse /xxA xA def /yyA yA def mark xB yB LPutVar { 1 index X check { exit } { /yyA ED /xxA ED } ifelse } loop /yyB ED /xxB ED cleartomark VPosEnd } def /HPutCurve { gsave newpath /SaveLPutVar /LPutVar load def LPutVar 8 -2 roll moveto curveto flattenpath /LPutVar [ {} {} {} {} pathforall ] cvx def grestore exec /LPutVar /SaveLPutVar load def } def /NCCoor { /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 xA1 yA1 ] cvx def /LPutPos { LPutVar LPutLine } def /HPutPos { LPutVar HPutLine } def /VPutPos { LPutVar VPutLine } def LPutVar } def /NCLine { NCCoor tx@Dict begin ArrowA CP 4 2 roll ArrowB lineto pop pop end } def /NCLines { false NArray n 0 eq { NCLine } { 2 copy yA sub exch xA sub Atan /AngleA ED n 2 mul dup index exch index yB sub exch xB sub Atan /AngleB ED GetEdgeA GetEdgeB /LPutVar [ xB1 yB1 n 2 mul 4 add 4 roll xA1 yA1 ] cvx def mark LPutVar tx@Dict begin false Line end /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } ifelse } def /NCCurve { GetEdgeA GetEdgeB xA1 xB1 sub yA1 yB1 sub Pyth 2 div dup 3 -1 roll mul /ArmA ED mul /ArmB ED /ArmTypeA 0 def /ArmTypeB 0 def GetArmA GetArmB xA2 yA2 xA1 yA1 tx@Dict begin ArrowA end xB2 yB2 xB1 yB1 tx@Dict begin ArrowB end curveto /LPutVar [ xA1 yA1 xA2 yA2 xB2 yB2 xB1 yB1 ] cvx def /LPutPos { t LPutVar BezierMidpoint } def /HPutPos { { HPutLines } HPutCurve } def /VPutPos { { VPutLines } HPutCurve } def } def /NCAngles { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def xA2 yA2 mtrx transform pop xB2 yB2 mtrx transform exch pop mtrx itransform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCAngle { GetEdgeA GetEdgeB GetArmB /mtrx AngleA matrix rotate def xB2 yB2 mtrx itransform pop xA1 yA1 mtrx itransform exch pop mtrx transform /y0 ED /x0 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 x0 y0 xA1 yA1 tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 x0 y0 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCBar { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def xA2 yA2 mtrx itransform pop xB2 yB2 mtrx itransform pop sub dup 0 mtrx transform 3 -1 roll 0 gt { /yB2 exch yB2 add def /xB2 exch xB2 add def } { /yA2 exch neg yA2 add def /xA2 exch neg xA2 add def } ifelse mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCDiag { GetEdgeA GetEdgeB GetArmA GetArmB mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCDiagg { GetEdgeA GetArmA yB yA2 sub xB xA2 sub Atan 180 add /AngleB ED GetEdgeB mark xB1 yB1 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def /NCLoop { GetEdgeA GetEdgeB GetArmA GetArmB /mtrx AngleA matrix rotate def xA2 yA2 mtrx transform loopsize add /yA3 ED /xA3 ED /xB3 xB2 yB2 mtrx transform pop def xB3 yA3 mtrx itransform /yB3 ED /xB3 ED xA3 yA3 mtrx itransform /yA3 ED /xA3 ED mark ArmB 0 ne { xB1 yB1 } if xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 ArmA 0 ne { xA1 yA1 } if tx@Dict begin false Line end /LPutVar [ xB1 yB1 xB2 yB2 xB3 yB3 xA3 yA3 xA2 yA2 xA1 yA1 ] cvx def /LPutPos { LPutLines } def /HPutPos { HPutLines } def /VPutPos { VPutLines } def } def % DG/SR modification begin - May 9, 1997 - Patch 1 %/NCCircle { 0 0 NodesepA nodeA \tx@GetEdge pop xA sub 2 div dup 2 exp r %r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add %exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 %mul add dup 5 1 roll 90 sub \tx@PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED /NCCircle { NodeSepA 0 NodeA 0 GetEdge pop 2 div dup 2 exp r r mul sub abs sqrt atan 2 mul /a ED r AngleA 90 add PtoC yA add exch xA add exch 2 copy /LPutVar [ 4 2 roll r AngleA ] cvx def /LPutPos { LPutVar t 360 mul add dup 5 1 roll 90 sub PtoC 3 -1 roll add /Y ED add /X ED /NAngle ED % DG/SR modification end } def /HPutPos { LPutPos } def /VPutPos { LPutPos } def r AngleA 90 sub a add AngleA 270 add a sub tx@Dict begin /angleB ED /angleA ED /r ED /c 57.2957 r Div def /y ED /x ED } def /NCBox { /d ED /h ED /AngleB yB yA sub xB xA sub Atan def /AngleA AngleB 180 add def GetEdgeA GetEdgeB /dx d AngleB sin mul def /dy d AngleB cos mul neg def /hx h AngleB sin mul neg def /hy h AngleB cos mul def /LPutVar [ xA1 hx add yA1 hy add xB1 hx add yB1 hy add xB1 dx add yB1 dy add xA1 dx add yA1 dy add ] cvx def /LPutPos { LPutLines } def /HPutPos { xB yB xA yA LPutLine } def /VPutPos { HPutPos } def mark LPutVar tx@Dict begin false Polygon end } def /NCArcBox { /l ED neg /d ED /h ED /a ED /AngleA yB yA sub xB xA sub Atan def /AngleB AngleA 180 add def /tA AngleA a sub 90 add def /tB tA a 2 mul add def /r xB xA sub tA cos tB cos sub Div dup 0 eq { pop 1 } if def /x0 xA r tA cos mul add def /y0 yA r tA sin mul add def /c 57.2958 r div def /AngleA AngleA a sub 180 add def /AngleB AngleB a add 180 add def GetEdgeA GetEdgeB /AngleA tA 180 add yA yA1 sub xA xA1 sub Pyth c mul sub def /AngleB tB 180 add yB yB1 sub xB xB1 sub Pyth c mul add def l 0 eq { x0 y0 r h add AngleA AngleB arc x0 y0 r d add AngleB AngleA arcn } { x0 y0 translate /tA AngleA l c mul add def /tB AngleB l c mul sub def 0 0 r h add tA tB arc r h add AngleB PtoC r d add AngleB PtoC 2 copy 6 2 roll l arcto 4 { pop } repeat r d add tB PtoC l arcto 4 { pop } repeat 0 0 r d add tB tA arcn r d add AngleA PtoC r h add AngleA PtoC 2 copy 6 2 roll l arcto 4 { pop } repeat r h add tA PtoC l arcto 4 { pop } repeat } ifelse closepath /LPutVar [ x0 y0 r AngleA AngleB h d ] cvx def /LPutPos { LPutVar /d ED /h ED /AngleB ED /AngleA ED /r ED /y0 ED /x0 ED t 1 le { r h add AngleA 1 t sub mul AngleB t mul add dup 90 add /NAngle ED PtoC } { t 2 lt { /NAngle AngleB 180 add def r 2 t sub h mul t 1 sub d mul add add AngleB PtoC } { t 3 lt { r d add AngleB 3 t sub mul AngleA 2 t sub mul add dup 90 sub /NAngle ED PtoC } { /NAngle AngleA 180 add def r 4 t sub d mul t 3 sub h mul add add AngleA PtoC } ifelse } ifelse } ifelse y0 add /Y ED x0 add /X ED } def /HPutPos { LPutPos } def /VPutPos { LPutPos } def } def /Tfan { /AngleA yB yA sub xB xA sub Atan def GetEdgeA w xA1 xB sub yA1 yB sub Pyth Pyth w Div CLW 2 div mul 2 div dup AngleA sin mul yA1 add /yA1 ED AngleA cos mul xA1 add /xA1 ED /LPutVar [ xA1 yA1 m { xB w add yB xB w sub yB } { xB yB w sub xB yB w add } ifelse xA1 yA1 ] cvx def /LPutPos { LPutLines } def /VPutPos@ { LPutVar flag { 8 4 roll pop pop pop pop } { pop pop pop pop 4 2 roll } ifelse } def /VPutPos { VPutPos@ VPutLine } def /HPutPos { VPutPos@ HPutLine } def mark LPutVar tx@Dict begin /ArrowA { moveto } def /ArrowB { } def false Line closepath end } def /LPutCoor { NAngle tx@Dict begin /NAngle ED end gsave CM STV CP Y sub neg exch X sub neg exch moveto setmatrix CP grestore } def /LPut { tx@NodeDict /LPutPos known { LPutPos } { CP /Y ED /X ED /NAngle 0 def } ifelse LPutCoor } def /HPutAdjust { Sin Cos mul 0 eq { 0 } { d Cos mul Sin div flag not { neg } if h Cos mul Sin div flag { neg } if 2 copy gt { pop } { exch pop } ifelse } ifelse s add flag { r add neg } { l add } ifelse X add /X ED } def /VPutAdjust { Sin Cos mul 0 eq { 0 } { l Sin mul Cos div flag { neg } if r Sin mul Cos div flag not { neg } if 2 copy gt { pop } { exch pop } ifelse } ifelse s add flag { d add } { h add neg } ifelse Y add /Y ED } def end % END pst-node.pro %%EndProcSet %%BeginProcSet: special.pro %! 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Fh(n)1358 1697 y Fn(x)25 b Ft(=)g Fn(x)31 b Ft(for)f(some)h Fn(n)24 b(>)h Ft(0)15 b Fi(g)32 b Ft(\(the)f(set)g(of)f(p)s(erio)s(dic)e (elemen)m(ts\);)164 1805 y Fs({)41 b Fn(R)327 1772 y Fh(\033)399 1805 y Ft(=)25 b Fi(f)15 b Fn(x)26 b Fi(2)f Fn(R)31 b Ft(suc)m(h)f(that)h Fn(\033)s(x)25 b Ft(=)g Fn(ux)31 b Ft(for)f(some)g Fn(u)c Fi(2)f Fn(R)2184 1772 y Fh(?)2238 1805 y Fi(g)31 b Ft(\(the)g(set)f(of)h(semi-in)m(v)-5 b(arian)m(t)29 b(elemen)m(ts\);)164 1916 y Fs({)41 b Fn(R)327 1883 y Fh(\033)369 1859 y Fc(?)454 1916 y Ft(=)j Fi(f)15 b Fn(x)45 b Fi(2)f Fn(R)31 b Ft(suc)m(h)f(that)h Fn(\033)1388 1883 y Fh(n)1435 1916 y Fn(x)45 b Ft(=)f Fn(ux)30 b Ft(for)g(some)h Fn(n)44 b(>)g Ft(0)p Fn(;)57 b(u)45 b Fi(2)f Fn(R)2761 1883 y Fh(?)2815 1916 y Fi(g)e Ft(\(the)h(set)f(of)g(semi-p)s(erio)s(dic)257 2024 y(elemen)m(ts\).)0 2167 y(It)f(is)f(clear)h(that)g(w)m(e)h(ha)m(v)m(e)g(the)f(inclusion)d Fn(R)1637 2181 y Fh(\033)1726 2167 y Fi(\022)43 b Fn(R)1910 2134 y Fh(\033)1999 2167 y Fi(\022)g Fn(R)2183 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Fn(ct)3020 4011 y Fh(m)3113 4044 y Fi(j)26 b Fn(c)f Fi(2)g Fn(k)s(;)48 b(m)25 b Fi(\025)g Ft(0)15 b Fi(g)p Fo(.)100 4185 y Ft(F)-8 b(or)28 b(example,)f(in)f Fn(C)7 b Ft([)p Fn(n)p Ft(],)28 b(let)f Fn(\033)j Ft(b)s(e)c(suc)m(h)h(that)h Fn(\033)s(n)d Ft(=)g Fn(q)s(n)h Ft(for)h(some)h Fn(q)g Fi(2)d Fn(C)2641 4152 y Fh(?)2680 4185 y Ft(.)39 b(The)27 b(prop)s(ert)m(y)f(holds)g (whenev)m(er)0 4293 y Fn(q)35 b Ft(is)c(not)h(a)g(ro)s(ot)h(of)f(unit)m (y)-8 b(.)45 b(Or)31 b(w)m(e)i(can)f(consider)f Fn(C)7 b Ft([)p Fn(n;)15 b(t)p Ft(],)32 b(with)f Fn(\033)k Ft(suc)m(h)d(that)g Fn(\033)2839 4312 y Fg(j)p Fh(C)2946 4293 y Ft(=)c Fn(id)3123 4307 y Fh(C)3182 4293 y Ft(,)65 b Fn(\033)s(n)28 b Ft(=)f Fn(n)21 b Ft(+)g(1)33 b(and)0 4405 y Fn(\033)s(t)25 b Ft(=)g(\()p Fn(n)20 b Ft(+)g(1\))p Fn(t)p Ft(;)31 b(in)f(other)g(w)m (ords)g Fn(t)g Ft(represen)m(ts)g Fn(n)p Ft(!.)1650 4670 y(4.)47 b Fs(Disp)s(ersion)0 4832 y(De\014nition)33 b(2.)40 b Ft(Let)29 b Fn(K)35 b Ft(b)s(e)27 b(a)i(\014eld)e(of)h(c)m (haracteristic)h(0.)41 b(Let)29 b Fn(\036)c Ft(:)h Fn(K)7 b Ft([)p Fn(X)g Ft(])26 b Fi(!)f Fn(K)7 b Ft([)p Fn(X)g Ft(])29 b(b)s(e)e(a)i(function.)39 b(Let)29 b Fn(p)f Ft(and)0 4940 y Fn(q)33 b Ft(b)s(e)d(non-zero)h(p)s(olynomials)c(in)i Fn(K)7 b Ft([)p Fn(X)g Ft(].)42 b(One)30 b(de\014nes)164 5081 y Fs({)41 b Ft(the)31 b(spread)f(of)g Fn(p)g Ft(and)g Fn(q)j Ft(with)c(resp)s(ect)h(to)i Fn(\036)p Ft(:)635 5255 y(Spr)773 5270 y Fh(\036)818 5255 y Ft(\(p)p Fn(;)15 b Ft(q\))25 b(=)g Fi(f)15 b Fn(m)26 b Fi(\025)f Ft(0)31 b(suc)m(h)f(that)h Fn(p)f Ft(and)g Fn(\036)2195 5217 y Fh(m)2261 5255 y Fn(q)k Ft(ha)m(v)m(e)d(a)g(non)f(trivial)e(gcd)16 b Fi(g)p eop %%Page: 67 3 67 2 bop 2541 66 a Ff(M.)23 b(Bronstein,)g(summary)e(b)n(y)j(A.)f(F)-6 b(redet)142 b Fq(67)164 266 y Fs({)41 b Ft(the)31 b(disp)s(ersion)c(of) j Fn(p)g Ft(and)g Fn(q)j Ft(with)c(resp)s(ect)i(to)g Fn(\036)p Ft(:)528 526 y(Dis)659 541 y Fh(\036)704 526 y Ft(\(p)p Fn(;)15 b Ft(q\))26 b(=)1035 340 y Fe(8)1035 422 y(<)1035 585 y(:)1157 417 y Fi(\000)p Ft(1)575 b(if)45 b(Spr)2085 432 y Fh(\036)2129 417 y Ft(\(p)p Fn(;)15 b Ft(q\))46 b(is)30 b(empt)m(y;)1157 525 y(max)q(\(Spr)o(\()p Fn(p;)15 b(q)s Ft(\)\))114 b(if)45 b(Spr)2085 540 y Fh(\036)2129 525 y Ft(\()p Fn(p;)15 b(q)s Ft(\))31 b(is)f(a)g(\014nite)g(nonempt)m (y)g(set;)1157 633 y(+)p Fi(1)529 b Ft(if)45 b(Spr)2085 648 y Fh(\036)2129 633 y Ft(\(p)p Fn(;)15 b Ft(q\))46 b(is)30 b(an)g(in\014nite)e(set.)0 792 y(These)i(de\014nitions)e(are)j (sp)s(ecialized)d(to)j(the)g(case)g Fn(p)25 b Ft(=)g Fn(q)s Ft(:)41 b(Spr)2182 807 y Fh(\036)2227 792 y Ft(\()p Fn(p)p Ft(\))25 b(=)g(Spr)2602 807 y Fh(\036)2647 792 y Ft(\()p Fn(p;)15 b(p)p Ft(\))31 b(and)f(Dis)3188 807 y Fh(\036)3233 792 y Ft(\()p Fn(p)p Ft(\))c(=)f(Dis)3602 807 y Fh(\036)3648 792 y Ft(\()p Fn(p;)15 b(p)p Ft(\).)100 924 y(Examples)29 b(are:)164 1056 y Fs({)41 b Ft(Dis)388 1075 y Fh(d=dx)540 982 y Fe(\000)581 1056 y Fn(p)p Ft(\()p Fn(x)p Ft(\))749 982 y Fe(\001)822 1056 y Ft(is)29 b(the)i(maxim)m(um)e (of)h(the)h(m)m(ultiplicit)m(y)c(of)k(a)g(ro)s(ot)f(of)h Fn(p)f Ft(min)m(us)f(1;)164 1177 y Fs({)41 b Ft(Spr)394 1199 y Fh(n)p Fg(!)p Fh(n)p Fq(+1)644 1103 y Fe(\000)686 1177 y Fn(p)p Ft(\()p Fn(n)p Ft(\))857 1103 y Fe(\001)929 1177 y Ft(is)30 b(\014nite)f(\(and)h(then)g(Dis)1803 1191 y Fh(n)p Fg(!)p Fh(n)p Fq(+1)2053 1103 y Fe(\000)2095 1177 y Fn(p)p Ft(\()p Fn(n)p Ft(\))2266 1103 y Fe(\001)2333 1177 y Fn(<)25 b Ft(+)p Fi(1)p Ft(\);)164 1285 y Fs({)41 b Ft(Dis)388 1299 y Fh(n)p Fg(!)p Fh(q)r(n)582 1285 y Ft(\()p Fn(n)p Ft(\))31 b(is)e(in\014nite.)0 1417 y(Let)j Fn(\033)i Ft(b)s(e)c(an)h(automorphism)e(of)j Fn(k)s Ft([)p Fn(t)p Ft(])f(suc)m(h)g(that)g Fn(\033)s(k)f Fi(\022)c Fn(k)s Ft(.)43 b(Then)30 b(the)h(disp)s(ersion)d(Dis)3077 1431 y Fh(\033)3123 1417 y Ft(\()p Fn(q)s Ft(\))k(is)e(in\014nite)f(if) h(and)0 1528 y(only)h(if)g(there)h(exists)g Fn(p)g Ft(in)e Fn(k)s Ft([)p Fn(t)p Ft(])1091 1495 y Fh(\033)1133 1471 y Fc(?)1175 1528 y Fi(n)p Fn(k)35 b Ft(suc)m(h)d(that)h Fn(p)e Ft(divides)f Fn(q)s Ft(.)45 b(Also,)33 b(the)f(disp)s(ersion)d (Dis)3159 1542 y Fh(\033)3205 1528 y Ft(\()p Fn(h;)15 b(q)s Ft(\))33 b(is)e(in\014nite)f(if)0 1638 y(and)g(only)f(if)h(there) g(exists)g Fn(p)g Ft(in)f Fn(k)s Ft([)p Fn(t)p Ft(])1258 1605 y Fh(\033)1300 1582 y Fc(?)1342 1638 y Fi(n)p Fn(k)34 b Ft(suc)m(h)c(that)h Fn(p)f Ft(divides)e Fn(q)33 b Ft(and)d Fn(\033)2558 1605 y Fh(n)2605 1638 y Fn(p)g Ft(divides)e Fn(h)p Ft(.)0 1758 y Fo(Example.)41 b Ft(Let)29 b Fn(a)c Ft(=)g(2)p Fn(n)834 1725 y Fq(7)890 1758 y Ft(+)17 b(19)p Fn(n)1123 1725 y Fq(6)1179 1758 y Ft(+)g(63)p Fn(n)1412 1725 y Fq(5)1468 1758 y Ft(+)f(81)p Fn(n)1700 1725 y Fq(4)1757 1758 y Ft(+)g(27)p Fn(n)1989 1725 y Fq(3)2058 1758 y Ft(b)s(e)28 b(in)f Fl(Q)9 b Ft([)p Fn(n)p Ft(])34 b(and)28 b Fn(\036)h Ft(b)s(e)f(the)h(automorphism)d(of)j Fl(Q)9 b Ft([)p Fn(n)p Ft(])0 1866 y(o)m(v)m(er)32 b Fl(Q)45 b Ft(that)31 b(maps)f Fn(n)f Ft(to)j Fn(n)19 b Ft(+)h(1.)41 b(The)30 b(resultan)m(t)g(of)h Fn(a)f Ft(and)g Fn(\036)2185 1833 y Fh(m)2252 1866 y Fn(a)g Ft(is)707 2022 y(4)p Fn(m)832 1985 y Fq(19)907 2022 y Ft(\(2)p Fn(m)21 b Ft(+)f(5\))1259 1985 y Fq(3)1299 2022 y Ft(\(2)p Fn(m)h Ft(+)f(1\))1651 1985 y Fq(3)1691 2022 y Ft(\(2)p Fn(m)h Fi(\000)f Ft(1\))2043 1985 y Fq(3)2083 2022 y Ft(\(2)p Fn(m)h Fi(\000)f Ft(5\))2435 1985 y Fq(3)2475 2022 y Ft(\()p Fn(m)g Fi(\000)g Ft(3\))2781 1985 y Fq(9)2821 2022 y Ft(\()p Fn(m)h Ft(+)f(3\))3128 1985 y Fq(9)3168 2022 y Fn(;)0 2179 y Ft(implying)27 b(that)k(Spr)711 2194 y Fh(\036)756 2179 y Ft(\()p Fn(a)p Ft(\))26 b(=)f Fi(f)p Ft(0)p Fn(;)15 b Ft(3)p Fi(g)32 b Ft(and)e(Dis)1556 2194 y Fh(\036)1602 2179 y Ft(\()p Fn(a)p Ft(\))c(=)f(3)0 2362 y(4.1.)47 b Fs(Splitting)38 b(factorization.)46 b Ft(One)32 b(no)m(w)h(extends)g(the)h(splitting)c(factorization)k(of)f (p)s(olynomials)d(to)k(dif-)0 2469 y(ference)d(\014eld:)39 b(let)30 b Fn(q)k Ft(in)29 b Fn(k)s Ft([)p Fn(t)p Ft(])h(b)s(e)g (decomp)s(osed)g(in)m(to)h(t)m(w)m(o)g(factors)h Fn(q)c Ft(=)d Fn(q)2504 2483 y Fg(1)p 2578 2419 44 4 v 2578 2469 a Fn(q)33 b Ft(suc)m(h)d(that)164 2602 y Fs({)41 b Ft(the)31 b(gcd)g(of)f Fn(q)725 2616 y Fg(1)830 2602 y Ft(and)p 1006 2552 V 29 w Fn(q)k Ft(is)29 b(equal)h(to)h(1,)164 2709 y Fs({)41 b Ft(for)30 b(all)g(irreducible)d(factor)k Fn(p)f Ft(of)h Fn(q)s Ft(,)60 b Fn(p)30 b Ft(divides)e Fn(q)1965 2723 y Fg(1)2070 2709 y Ft(if)h Fn(p)h Ft(is)g(in)f Fn(k)s Ft([)p Fn(t)p Ft(])2560 2676 y Fh(\033)2602 2653 y Fc(?)2643 2709 y Ft(,)164 2820 y Fs({)41 b Ft(and)30 b(for)g(all)f(irreducible)e(factor)32 b Fn(p)e Ft(of)g Fn(q)s Ft(,)61 b Fn(p)30 b Ft(divides)p 2101 2770 V 28 w Fn(q)j Ft(if)d Fn(p)g Ft(is)f(not)i(in)e Fn(k)s Ft([)p Fn(t)p Ft(])2827 2787 y Fh(\033)2869 2764 y Fc(?)2910 2820 y Ft(.)0 2952 y(The)35 b(p)s(olynomial)d Fn(q)707 2966 y Fg(1)817 2952 y Ft(is)i(the)h Fo(in\014nite)i(p)-5 b(art)46 b Ft(of)35 b Fn(q)s Ft(,)i(and)p 1988 2902 V 34 w Fn(q)h Ft(is)c(its)h Fo(\014nite)i(p)-5 b(art)p Ft(.)57 b(W)-8 b(e)36 b(note)g(that)g(the)f(disp)s(ersion)0 3060 y(Dis)131 3074 y Fh(\033)177 3060 y Ft(\()p 212 3010 V Fn(q)t Ft(\))29 b(is)f(\014nite,)h(the)h(disp)s(ersion)25 b(Dis)1381 3074 y Fh(\033)1428 3060 y Ft(\()p Fn(q)1504 3074 y Fg(1)1579 3060 y Ft(\))k(is)f(in\014nite,)g(and)g(for)h(all)g Fn(h)g Ft(the)h(disp)s(ersion)c(Dis)3299 3074 y Fh(\033)3346 3060 y Ft(\()p Fn(h;)p 3473 3010 V 15 w(q)t Ft(\))j(is)g(\014nite.)0 3243 y(4.2.)47 b Fn(\033)s Fs(-Orbits.)e Ft(Giv)m(en)35 b Fn(\013)g Ft(and)g Fn(\014)40 b Ft(in)33 b(a)j(\014eld)d Fn(K)7 b Ft(,)36 b(the)f(problem)f(of)h(the)g(orbit)f(is)g(to)i(\014nd) d Fn(m)g Fi(\025)f Ft(0)k(suc)m(h)e(that)0 3351 y Fn(\013)58 3318 y Fh(m)161 3351 y Ft(=)j Fn(\014)5 b Ft(.)60 b(A)38 b(b)s(ound)d(for)h(the)i(smallest)e Fn(m)h Ft(suc)m(h)f(that)i Fn(\013)2058 3318 y Fh(m)2161 3351 y Ft(=)f Fn(\014)42 b Ft(is)36 b(giv)m(en)h(in)f([3].)61 b(The)37 b(main)f(ideas)g(are)i (as)0 3461 y(follo)m(ws:)48 b(if)34 b(there)h(exists)f Fn(d)h Ft(suc)m(h)f(that)h Fn(\013)1472 3428 y Fh(d)1545 3461 y Ft(=)d(1)j(then)f(one)h(can)g(test)h(whether)e Fn(\013)2874 3428 y Fh(i)2934 3461 y Ft(=)e Fn(\014)40 b Ft(for)35 b(0)d Fi(\024)g Fn(i)h Fi(\024)f Fn(d)p Ft(.)53 b(If)34 b(it)0 3569 y(is)i(not)g(the)h(case,)j(then)c(the)h(orbit)e (problem)g(has)i(no)f(solution,)h(otherwise)f(its)g(solutions)f (consist)i(of)f(all)g(the)0 3677 y(in)m(tegers)d(of)g(the)g(form)f Fn(i)852 3691 y Fq(0)913 3677 y Ft(+)22 b Fn(k)s(d)1103 3691 y Fq(0)1176 3677 y Ft(where)32 b Fn(k)g Fi(\025)d Ft(0,)34 b Fn(i)1755 3691 y Fq(0)1827 3677 y Ft(is)e(the)h(smallest)f Fn(i)d Fi(\025)g Ft(0)k(suc)m(h)g(that)g Fn(\013)3135 3644 y Fh(i)3193 3677 y Ft(=)c Fn(\014)37 b Ft(and)c Fn(d)3608 3691 y Fq(0)3680 3677 y Ft(is)f(the)0 3785 y(smallest)e Fn(d)c(>)f Ft(0)31 b(suc)m(h)f(that)h Fn(\013)1055 3752 y Fh(d)1122 3785 y Ft(=)25 b(1.)41 b(One)30 b(can)h(no)m(w)g (assume)f(that)h Fn(\013)g Ft(is)f(not)h(a)g(ro)s(ot)f(of)h(unit)m(y)-8 b(,)30 b(whic)m(h)g(implies)0 3893 y(that)i(the)g(orbit)f(problem)f (has)h(at)i(most)f(one)g(solution.)43 b(If)31 b Fn(\013)h Ft(is)f(transcenden)m(tal)g(o)m(v)m(er)i Fl(Q)9 b Ft(,)38 b(the)32 b(orbit)f(problem)0 4001 y(has)j(a)h(solution)e(if)h(and)f (only)h(if)f Fn(\014)40 b Ft(is)34 b(algebraic)g(o)m(v)m(er)i Fl(Q)8 b Ft(\()q Fn(\013)p Ft(\).)59 b(Lo)s(oking)34 b(at)i(the)e(degree)h(at)h(whic)m(h)d Fn(\013)i Ft(app)s(ears)0 4109 y(in)g Fn(\014)41 b Ft(giv)m(es)36 b(at)g(most)h(one)f(candidate)f (solution)g(for)g(the)h(orbit)f(problem.)56 b(One)35 b(can)i(no)m(w)f(assume)f(that)i Fn(\013)f Ft(is)0 4217 y(algebraic)26 b(o)m(v)m(er)h Fl(Q)8 b Ft(.)46 b(This)24 b(generalizes)h(to)i(\014nd)d Fn(m)h Fi(\025)g Ft(0)i(suc)m(h)e(that)i Fn(\013)2366 4184 y Fh(m;\033)2520 4217 y Ft(=)e Fn(\013)p Ft(\()p Fn(\033)s(\013)p Ft(\))15 b Fn(:)g(:)g(:)j Ft(\()p Fn(\033)3085 4184 y Fh(m)p Fg(\000)p Fq(1)3243 4217 y Fn(\013)p Ft(\))26 b(=)f Fn(\014)57 b Ft(\(see)26 b([3)q(]\).)0 4399 y(4.3.)47 b Fs(Computation)42 b(of)h(the)g(disp)s(ersion.)j Ft(Let)39 b Fn(\033)h Ft(:)e Fn(K)7 b Ft([)p Fn(X)g Ft(])38 b Fi(!)f Fn(K)7 b Ft([)p Fn(X)g Ft(])38 b(b)s(e)f(an)h(automorphism)e (suc)m(h)h(that)0 4507 y Fn(\033)s(K)32 b Fi(\022)25 b Fn(K)7 b Ft(.)41 b(Then)211 4708 y(Spr)349 4722 y Fh(\033)394 4580 y Fe(\022)461 4622 y(Y)507 4817 y Fh(i)593 4708 y Fn(p)639 4667 y Fh(e)672 4677 y Fc(i)639 4735 y Fh(i)702 4708 y Fn(;)742 4622 y Fe(Y)784 4817 y Fh(j)873 4708 y Fn(q)917 4657 y Fh(f)951 4667 y Fc(j)914 4735 y Fh(j)988 4580 y Fe(\023)1080 4708 y Ft(=)1176 4622 y Fe([)1188 4817 y Fh(i;j)1292 4708 y Ft(Spr)1430 4722 y Fh(\033)1475 4708 y Ft(\()p Fn(p)1556 4722 y Fh(i)1585 4708 y Fn(;)15 b(q)1666 4722 y Fh(j)1702 4708 y Ft(\))92 b(and)105 b(Dis)2212 4722 y Fh(\033)2258 4580 y Fe(\022)2325 4622 y(Y)2371 4817 y Fh(i)2457 4708 y Fn(p)2503 4667 y Fh(e)2536 4677 y Fc(i)2503 4735 y Fh(i)2566 4708 y Fn(;)2606 4622 y Fe(Y)2648 4817 y Fh(j)2737 4708 y Fn(q)2781 4657 y Fh(f)2815 4667 y Fc(j)2778 4735 y Fh(j)2852 4580 y Fe(\023)2944 4708 y Ft(=)25 b(max)3087 4767 y Fh(i;j)3224 4708 y Ft(Dis)3355 4722 y Fh(\033)3402 4708 y Ft(\()p Fn(p)3483 4722 y Fh(i)3511 4708 y Fn(;)15 b(q)3592 4722 y Fh(j)3628 4708 y Ft(\))p Fn(:)0 4959 y Ft(The)29 b(computation)h(of)f(the)h(disp)s(ersion)d (reduces)i(to)h(the)g(computation)g(of)f(the)h(disp)s(ersion)d(of)i(t)m (w)m(o)i(irreducible)0 5067 y(p)s(olynomials.)100 5175 y(Let)k Fn(p)g Ft(and)f Fn(q)j Ft(b)s(e)e(irreducible)c(p)s (olynomials.)52 b(Let)35 b Fn(m)g Ft(b)s(e)f(in)f(Spr)2403 5189 y Fh(\033)2449 5175 y Ft(\()p Fn(p;)15 b(q)s Ft(\).)55 b(This)33 b(means)i(that)g(the)g(greatest)0 5283 y(common)e(divisor)e (of)i Fn(p)g Ft(and)g Fn(\033)1085 5250 y Fh(m)1151 5283 y Fn(q)j Ft(is)c(not)i(trivial.)46 b(The)33 b(p)s(olynomials)d(b)s (eing)i(irreducible,)e(this)i(is)g(equiv)-5 b(alen)m(t)p eop %%Page: 68 4 68 3 bop 0 66 a Fq(68)142 b Ff(Di\013erence)24 b(Equations)h(with)e (Hyp)r(ergeometric)h(Co)r(e\016cien)n(ts)0 266 y Ft(to)32 b(the)g(existence)h(of)f Fn(u)f Ft(in)g Fn(K)1039 233 y Fh(?)1110 266 y Ft(suc)m(h)g(that)h Fn(\033)1569 233 y Fh(m)1636 266 y Fn(q)e Ft(=)e Fn(up)p Ft(.)44 b(This)30 b(implies)f(that)k(deg)16 b Fn(p)28 b Ft(=)f(deg)17 b Fn(q)s Ft(.)44 b(One)32 b(just)f(has)g(to)0 374 y(consider)e (irreducible)e(p)s(olynomials)h(with)h(common)i(degree.)100 484 y(Let)39 b Fn(p)g Ft(and)f Fn(q)k Ft(b)s(e)c(monic)g(irreducible)e (p)s(olynomials)g(of)j Fn(k)s Ft([)p Fn(t)p Ft(])g(with)f(degree)h Fn(n)p Ft(:)58 b Fn(p)39 b Ft(=)g Fn(t)3162 451 y Fh(n)3235 484 y Ft(+)3332 416 y Fe(P)3427 442 y Fh(n)p Fg(\000)p Fq(1)3427 511 y Fh(i)p Fq(=0)3580 484 y Fn(p)3626 498 y Fh(i)3654 484 y Fn(t)3687 451 y Fh(i)3754 484 y Ft(and)0 604 y Fn(q)48 b Ft(=)d Fn(t)238 571 y Fh(n)313 604 y Ft(+)412 536 y Fe(P)508 562 y Fh(n)p Fg(\000)p Fq(1)508 631 y Fh(i)p Fq(=0)660 604 y Fn(q)701 618 y Fh(i)729 604 y Fn(t)762 571 y Fh(i)790 604 y Ft(.)76 b(Assume)42 b(that)h Fn(\033)s(t)i Ft(=)g Fn(at)d Ft(for)h(some)f Fn(a)j Fi(2)g Fn(k)2461 571 y Fh(?)2501 604 y Ft(.)76 b(Then)42 b Fn(m)g Ft(is)f(in)g(Spr)3332 626 y Fh(\033)3379 604 y Ft(\()p Fn(p;)15 b(q)s Ft(\))43 b(implies)0 716 y Fn(\013)58 672 y Fh(m;\033)58 743 y(i)234 716 y Ft(=)j Fn(\014)402 730 y Fh(i)473 716 y Ft(for)d(all)f Fn(i)i Ft(suc)m(h)e(that)i Fn(p)1312 730 y Fh(i)1340 716 y Fn(q)1381 730 y Fh(i)1455 716 y Fi(6)p Ft(=)i(0,)h(where)42 b Fn(\014)2015 730 y Fh(i)2090 716 y Ft(=)47 b Fn(q)2249 730 y Fh(i)2276 716 y Fn(=p)2367 730 y Fh(i)2439 716 y Ft(and)42 b Fn(\013)2686 730 y Fh(i)2761 716 y Ft(=)k Fn(a)2926 683 y Fh(n)p Fg(\000)p Fh(i)3052 716 y Fn(q)3093 730 y Fh(i)3121 716 y Fn(=\033)s(q)3262 730 y Fh(i)3290 716 y Ft(.)79 b(Therefore,)46 b(if)0 824 y(Spr)138 838 y Fh(\033)183 824 y Ft(\()p Fn(p;)15 b(q)s Ft(\))36 b(is)e(not)h(empt)m(y)g(then)f Fn(p)1220 838 y Fh(i)1283 824 y Ft(and)g Fn(q)1505 838 y Fh(i)1567 824 y Ft(v)-5 b(anish)34 b(sim)m(ultaneously)-8 b(.)52 b(If)34 b Fn(p)e Ft(=)g Fn(q)k Ft(=)c Fn(t)j Ft(then)f(Dis\()p Fn(p;)15 b(q)s Ft(\))33 b(=)f(+)p Fi(1)p Ft(.)0 932 y(Otherwise,)g (this)f(reduces)h(to)i(the)e(orbit)g(problem)f Fn(\013)1874 899 y Fh(m;\033)2032 932 y Ft(=)e Fn(\014)37 b Ft(for)32 b Fn(\013;)15 b(\014)39 b Ft(in)31 b Fn(k)2707 899 y Fh(?)2779 932 y Ft(and)h Fn(m)g Ft(in)g(Spr)n(\()p Fn(p;)15 b(q)s Ft(\).)48 b(Remark)0 1042 y(that)31 b(if)f Fn(\033)s(w)e Fi(6)p Ft(=)d Fn(a)573 1009 y Fh(d)613 1042 y Fn(w)33 b Ft(for)e(all)e Fn(w)k Ft(in)c Fn(k)1231 1009 y Fh(?)1301 1042 y Ft(and)h Fn(d)c(>)f Ft(0)31 b(then)f Fn(\013)1988 1009 y Fh(d)2054 1042 y Fi(6)p Ft(=)25 b(1)31 b(for)f(all)g Fn(d)c(>)f Ft(0.)41 b(So,)31 b(the)f(orbit)g(problem)f(has)h(at)0 1150 y(most)h(one)f(solution)f(and)h(then)g(Spr)1254 1172 y Fh(\033)1301 1150 y Ft(\()p Fn(p;)15 b(q)s Ft(\))31 b(has)f(at)h(most)g(one)f(elemen)m(t.)100 1258 y(One)44 b(can)h(extend)f(the)h(computation)f(of)h(the)g(disp)s(ersion)c(to)k (rational)f(functions:)68 b(let)44 b Fn(f)58 b Ft(=)49 b Fn(p=q)e Ft(with)0 1366 y(relativ)m(ely)36 b(prime)f Fn(p)i Ft(and)f Fn(q)j Ft(in)d Fn(C)7 b Ft([)p Fn(n)p Ft(].)59 b(Let)37 b(Dis)1687 1380 y Fh(\033)1734 1366 y Ft(\()p Fn(f)10 b Ft(\))36 b(=)f(max)2170 1293 y Fe(\000)2212 1366 y Ft(Dis)2343 1380 y Fh(\033)2389 1366 y Ft(\()p Fn(p)p Ft(\))p Fn(;)15 b Ft(Dis)2676 1380 y Fh(\033)2724 1366 y Ft(\()p Fn(p;)g(q)s Ft(\))p Fn(;)g Ft(Dis)3095 1380 y Fh(\033)3142 1366 y Ft(\()p Fn(q)s(;)g(p)p Ft(\))p Fn(;)g Ft(Dis)3513 1380 y Fh(\033)3561 1366 y Ft(\()p Fn(q)s Ft(\))3675 1293 y Fe(\001)3754 1366 y Ft(and)0 1475 y Fn(\027)45 1489 y Fg(1)120 1475 y Ft(\()p Fn(f)10 b Ft(\))44 b(=)f(deg)17 b Fn(q)31 b Fi(\000)c Ft(deg)17 b Fn(p)p Ft(.)74 b(Then)41 b Fn(\027)1318 1489 y Fg(1)1392 1475 y Ft(\()p Fn(f)1482 1442 y Fh(m;\033)1611 1475 y Ft(\))j(=)g Fn(m\027)1930 1489 y Fg(1)2004 1475 y Ft(\()p Fn(f)10 b Ft(\).)75 b(And)40 b(if)h Fn(f)51 b Ft(is)40 b(not)i(in)f Fn(C)48 b Ft(then)41 b(Dis)3485 1489 y Fh(\033)3531 1475 y Ft(\()p Fn(f)3621 1442 y Fh(m;\033)3750 1475 y Ft(\))j(=)0 1583 y(Dis)131 1597 y Fh(\033)177 1583 y Ft(\()p Fn(f)10 b Ft(\))21 b(+)f Fn(m)g Fi(\000)f Ft(1.)100 1691 y(This)28 b(last)i(equalit)m(y)g(allo)m(ws)g(us)f(to)i(reduce)f (orbit)g(problems)e(to)j(disp)s(ersions)c(whenev)m(er)j Fn(\013)h Ft(is)e(not)h(constan)m(t.)918 1884 y(5.)47 b Fs(Rational)35 b(Solutions)g(of)g(Di\013erence)h(Equations)100 2046 y Ft(Let)30 b Fn(t)f Ft(b)s(e)g(a)h(monomial)e(o)m(v)m(er)j Fn(k)e Ft(=)c Fn(C)7 b Ft(\()p Fn(n)p Ft(\).)40 b(Let)30 b Fn(\033)i Ft(b)s(e)d(suc)m(h)h(that)g Fn(\033)s(n)25 b Ft(=)g Fn(n)18 b Ft(+)g(1)30 b(and)f Fn(\033)s(t)c Ft(=)g Fn(at)30 b Ft(for)f(some)h Fn(a)g Ft(in)e Fn(k)0 2164 y Ft(suc)m(h)i(that)i Fn(\033)s(w)c Fi(6)p Ft(=)e Fn(a)696 2131 y Fh(d)737 2164 y Fn(w)r Ft(,)31 b(for)g(all)e Fn(w)k Ft(in)d Fn(k)1381 2131 y Fh(?)1451 2164 y Ft(and)g Fn(d)d(>)e Ft(0.)42 b(Let)32 b Fn(L)25 b Ft(=)2258 2096 y Fe(P)2354 2122 y Fh(N)2354 2191 y(i)p Fq(=0)2487 2164 y Fn(a)2535 2178 y Fh(i)2564 2164 y Fn(\033)2619 2131 y Fh(i)2678 2164 y Ft(b)s(e)30 b(a)h(linear)e(di\013erence)h(op)s (erator,)0 2272 y(with)e(the)h Fn(a)409 2286 y Fh(i)438 2272 y Ft('s)g(in)f Fn(k)s Ft([)p Fn(t)p Ft(])h(and)g(b)s(oth)f Fn(a)1232 2286 y Fq(0)1301 2272 y Ft(and)h Fn(a)1525 2286 y Fh(N)1621 2272 y Ft(not)g(equal)g(to)h(0.)41 b(Let)29 b Fn(b)h Ft(b)s(e)e(in)g Fn(k)s Ft([)p Fn(t)p Ft(].)41 b(The)28 b(aim)h(of)g(this)g(section)g(is)0 2380 y(to)i(describ)s(ed)d (an)j(algorithm)e(to)i(\014nd)e Fn(y)k Ft(in)c Fn(k)s Ft(\()p Fn(t)p Ft(\))i(suc)m(h)f(that)h Fn(L)p Ft(\()p Fn(y)s Ft(\))26 b(=)f Fn(b)30 b Ft(\(if)g(there)g(exists)g(suc)m(h)g(a) h Fn(y)s Ft(\).)0 2549 y(5.1.)47 b Fs(Denominator)41 b(of)g(a)h(rational)f(solution.)46 b Ft(The)36 b(\014rst)f(problem)g (is)g(to)i(\014nd)d(a)j(b)s(ound)d(for)i(the)g(\014nite)0 2657 y(part)31 b(of)h(an)m(y)f Fn(y)k Ft(in)30 b Fn(k)s Ft(\()p Fn(t)p Ft(\))i(suc)m(h)f(that)h Fn(L)p Ft(\()p Fn(y)s Ft(\))27 b(=)g Fn(b)p Ft(.)43 b(This)30 b(means)h(to)h(compute)g (a)g(p)s(olynomial)c Fn(q)34 b Ft(in)c Fn(k)s Ft([)p Fn(t)p Ft(])i(suc)m(h)f(that)0 2767 y(if)g Fn(L)p Ft(\()p Fn(y)s Ft(\))e(=)g Fn(b)j Ft(then)g Fn(y)s(q)g Ft(=)c Fn(p=d)1032 2781 y Fg(1)1140 2767 y Ft(where)k Fn(p)g Ft(is)f(in)g Fn(k)s Ft([)p Fn(t)p Ft(])i(and)f Fn(d)2076 2781 y Fg(1)2183 2767 y Ft(in)f Fn(k)s Ft([)p Fn(t)p Ft(])2424 2734 y Fh(\033)2466 2711 y Fc(?)2508 2767 y Ft(.)47 b(W)-8 b(e)33 b(outline)e(the)i(ideas)f(here,)h(pro)s(ofs)0 2875 y(and)d(tec)m(hnical)g(details)f(are)i(giv)m(en)g(in)e([5].)100 2983 y(Let)34 b Fn(a)314 2997 y Fq(0)388 2983 y Ft(b)s(e)g(decomp)s (osed:)48 b Fn(a)1115 2997 y Fq(0)1187 2983 y Ft(=)31 b Fn(a)1337 2997 y Fq(0)p Fh(;)p Fg(1)p 1467 2933 88 4 v 1467 2983 a Fn(a)1515 2997 y Fq(0)1555 2983 y Ft(.)52 b(Let)35 b Fn(y)i Ft(b)s(e)d(in)f Fn(k)s Ft(\()p Fn(t)p Ft(\))i(suc)m(h)f(that)h Fn(L)p Ft(\()p Fn(y)s Ft(\))d(=)g Fn(b)p Ft(,)k(where)d Fn(y)i Ft(=)d Fn(p=d)j Ft(and)0 3099 y Fn(d)26 b Ft(=)f Fn(d)216 3113 y Fg(1)p 291 3025 48 4 v 291 3099 a Fn(d)p Ft(.)40 b(Then)28 b(Dis)770 3113 y Fh(\033)817 3099 y Ft(\()p 852 3025 V Fn(d)q Ft(\))d Fi(\024)g Ft(max)1225 3026 y Fe(\000)1267 3099 y Fi(\000)p Ft(1)p Fn(;)15 b Ft(Dis)1554 3113 y Fh(\033)1601 3099 y Ft(\()p Fn(a)1684 3113 y Fh(N)1752 3099 y Fn(;)p 1792 3049 88 4 v 15 w(a)1840 3113 y Fq(0)1880 3099 y Ft(\))j Fi(\000)g Fn(N)2105 3026 y Fe(\001)2146 3099 y Ft(.)41 b(Let)29 b Fn(h)d(>)f Ft(0)30 b(b)s(e)e(an)h(in)m(teger.)41 b(One)29 b(can)h(compute)0 3222 y(an)d(op)s(erator)g Fn(L)548 3237 y Fh(h)618 3222 y Ft(=)e Fn(b)753 3236 y Fh(s)790 3222 y Fn(\033)845 3189 y Fh(sh)936 3222 y Ft(+)14 b Fn(b)1060 3236 y Fh(s)p Fg(\000)p Fq(1)1186 3222 y Fn(\033)1241 3189 y Fq(\()p Fh(s)p Fg(\000)p Fq(1\))p Fh(h)1478 3222 y Ft(+)g Fi(\001)h(\001)g(\001)e Ft(+)h Fn(b)1805 3236 y Fq(0)1871 3222 y Ft(suc)m(h)27 b(that)h Fn(L)2329 3237 y Fh(h)2398 3222 y Ft(=)d Fn(R)q(L)i Ft(for)g(some)g Fn(R)h Ft(in)e Fn(k)s Ft(\()p Fn(t)p Ft(\)[)p Fn(\033)s Ft(].)41 b(It)27 b(follo)m(ws)0 3330 y(that)h Fn(L)256 3345 y Fh(h)300 3330 y Ft(\()p Fn(y)s Ft(\))e(=)f Fn(R)q(b)i Ft(for)f(an)m(y)i Fn(b)f Ft(in)e Fn(k)s Ft([)p Fn(t)p Ft(])j(and)e(an)m(y)i(solution)d Fn(y)30 b Ft(in)c Fn(k)s Ft(\()p Fn(t)p Ft(\))h(of)g Fn(L)p Ft(\()p Fn(y)s Ft(\))f(=)f Fn(b)p Ft(.)40 b(W)-8 b(e)28 b(get)g(that)g(ev)m(ery)f(solution)0 3438 y Fn(y)33 b Ft(in)c Fn(k)s Ft(\()p Fn(t)p Ft(\))i(of)g Fn(L)p Ft(\()p Fn(y)s Ft(\))25 b(=)g Fn(b)31 b Ft(satis\014es)f(an)g (equation)g(of)h(the)f(form)1367 3594 y Fn(c)1406 3608 y Fh(s)1443 3594 y Fn(\033)1498 3557 y Fh(hs)1576 3594 y Ft(\()p Fn(y)s Ft(\))20 b(+)g Fi(\001)15 b(\001)g(\001)22 b Ft(+)d Fn(c)2061 3608 y Fq(1)2101 3594 y Fn(\033)2156 3557 y Fh(h)2201 3594 y Ft(\()p Fn(y)s Ft(\))26 b(=)f Fn(d)2488 3609 y Fh(h)0 3748 y Ft(where)34 b Fn(c)306 3762 y Fq(0)346 3748 y Fn(;)15 b(:)g(:)g(:)i(;)e(c)587 3762 y Fh(s)624 3748 y Fn(;)g(d)711 3763 y Fh(h)792 3748 y Ft(are)35 b(in)e Fn(k)s Ft([)p Fn(t)p Ft(])i(and)g Fn(c)1447 3762 y Fh(s)1516 3748 y Fi(6)p Ft(=)e(0.)54 b(If)35 b Fn(h)g Ft(w)m(as)g(c)m(hosen)g(suc)m(h)g(that)g(Dis)2943 3762 y Fh(\033)2990 3748 y Ft(\()p 3025 3674 48 4 v Fn(d)p Ft(\))e Fn(<)g(h)i Ft(then)p 3542 3674 V 34 w Fn(d)g Ft(divides)0 3858 y(gcd)136 3880 y Fq(0)p Fg(\024)p Fh(i)p Fg(\024)p Fh(s)342 3858 y Ft(\()p Fn(\033)432 3825 y Fg(\000)p Fh(ih)557 3858 y Fn(c)596 3872 y Fh(i)624 3858 y Ft(\).)40 b(This)27 b(giv)m(es)h(us)g(a)g(p)s(olynomial)e Fn(q)31 b Ft(suc)m(h)c(that)i(if)e Fn(L)p Ft(\()p Fn(y)s Ft(\))f(=)f Fn(b)j Ft(then)g Fn(q)s(y)g Ft(=)d Fn(p=d)3281 3872 y Fg(1)3384 3858 y Ft(with)i Fn(p)h Ft(in)e Fn(k)s Ft([)p Fn(t)p Ft(])0 3979 y(and)k Fn(d)224 3993 y Fg(1)329 3979 y Ft(in)f Fn(k)s Ft([)p Fn(t)p Ft(])568 3946 y Fh(\033)610 3922 y Fc(?)0 4097 y Fo(Example.)43 b Ft(Consider)29 b Fn(y)s Ft(\()p Fn(n)20 b Ft(+)g(2\))h Fi(\000)f Ft(\()p Fn(n)p Ft(!)h(+)f Fn(n)p Ft(\))p Fn(y)s Ft(\()p Fn(n)g Ft(+)g(1\))i(+)e Fn(n)p Ft(\()p Fn(n)p Ft(!)g Fi(\000)g Ft(1\))p Fn(y)s Ft(\()p Fn(n)p Ft(\))27 b(=)e(0.)42 b(If)31 b(w)m(e)g(de\014ne)f Fn(\033)k Ft(b)m(y)c Fn(\033)s(n)c Ft(=)f Fn(n)20 b Ft(+)h(1)0 4205 y(and)37 b Fn(\033)s(t)h Ft(=)f(\()p Fn(n)25 b Ft(+)g(1\))p Fn(t)38 b Ft(then)f(the)h(asso)s (ciated)g(di\013erence)g(op)s(erator)g(is)e Fn(\033)2540 4172 y Fq(2)2605 4205 y Fi(\000)25 b Ft(\()p Fn(t)g Ft(+)g Fn(n)p Ft(\))p Fn(\033)j Ft(+)d Fn(n)p Ft(\()p Fn(t)g Fi(\000)g Ft(1\).)63 b Fn(a)3616 4219 y Fh(N)3721 4205 y Ft(=)37 b(1,)0 4317 y Fn(a)48 4331 y Fq(0)117 4317 y Ft(=)p 217 4267 88 4 v 29 w Fn(a)265 4331 y Fq(0)334 4317 y Ft(=)29 b Fn(n)p Ft(\()p Fn(t)22 b Fi(\000)f Ft(1\))34 b(and)e(Dis)1095 4331 y Fh(\033)1141 4317 y Ft(\()p Fn(a)1224 4331 y Fh(N)1292 4317 y Fn(;)p 1332 4267 V 15 w(a)1380 4331 y Fq(0)1420 4317 y Ft(\))e(=)f Fi(\000)p Ft(1.)48 b(Then)32 b(Dis)2145 4331 y Fh(\033)2191 4317 y Ft(\()p 2226 4243 48 4 v Fn(d)q Ft(\))e Fi(\024)f(\000)p Ft(1)k(and)p 2767 4243 V 32 w Fn(d)d Fi(2)f Fn(C)7 b Ft(\()p Fn(n)p Ft(\).)48 b(So,)33 b(if)f(there)h(exists)0 4425 y Fn(y)28 b Fi(2)d Fn(C)7 b Ft(\()p Fn(n)p Ft(\)\()p Fn(t)p Ft(\))31 b(suc)m(h)f(that)h Fn(L)p Ft(\()p Fn(y)s Ft(\))25 b(=)g Fn(b)30 b Ft(then)h Fn(y)i Ft(is)c(in)g Fn(C)7 b Ft(\()p Fn(n)p Ft(\)[)p Fn(t;)15 b(t)2073 4392 y Fg(\000)p Fq(1)2168 4425 y Ft(].)0 4543 y Fo(R)-5 b(emark.)43 b Ft(The)30 b(same)g(results)f(holds)g(for)h(the)h Fn(q)s Ft(-di\013erence)f (equation:)40 b(let)30 b Fn(q)k Ft(b)s(e)29 b(transcenden)m(tal)h(o)m (v)m(er)i Fl(Q)9 b Ft(.)46 b(Let)0 4651 y Fn(\033)34 b Ft(b)s(e)29 b(suc)m(h)h(that)h Fn(\033)s(x)26 b Ft(=)f Fn(q)s(x)p Ft(.)40 b(Consider)29 b(the)h Fn(q)s Ft(-di\013erence)g (equation)807 4800 y Fn(q)851 4763 y Fq(3)891 4800 y Ft(\()p Fn(q)s(x)20 b Ft(+)g(1\))p Fn(y)s Ft(\()p Fn(q)1340 4763 y Fq(2)1380 4800 y Fn(x)p Ft(\))h Fi(\000)e Ft(2)p Fn(q)1667 4763 y Fq(2)1707 4800 y Ft(\()p Fn(x)i Ft(+)f(1\))p Fn(y)s Ft(\()p Fn(q)s(x)p Ft(\))h(+)f(\()p Fn(x)g Ft(+)g Fn(q)s Ft(\))p Fn(y)s Ft(\()p Fn(x)p Ft(\))26 b(=)f(0)-2926 b(\(1\))0 4950 y(W)-8 b(e)30 b(ha)m(v)m(e)p 363 4900 88 4 v 30 w Fn(a)411 4964 y Fq(0)476 4950 y Ft(=)25 b Fn(x)16 b Ft(+)h Fn(q)s Ft(,)29 b Fn(a)874 4964 y Fq(2)939 4950 y Ft(=)c Fn(q)1079 4917 y Fq(3)1118 4950 y Ft(\()p Fn(q)s(x)17 b Ft(+)g(1\).)41 b(The)28 b(resultan)m(t)g(of)h Fn(a)2215 4964 y Fq(2)2283 4950 y Ft(and)f Fn(\033)2513 4917 y Fh(m)2580 4950 y Ft(\()p 2615 4900 V Fn(a)2663 4964 y Fq(0)2703 4950 y Ft(\))h(is)f Fn(q)2901 4917 y Fq(3)2940 4950 y Ft(\()p Fn(q)3019 4917 y Fq(2)3076 4950 y Fi(\000)16 b Fn(q)3207 4917 y Fh(m)3274 4950 y Ft(\),)29 b(whic)m(h)f(implies)0 5062 y(that)41 b(Dis)338 5076 y Fh(\033)384 5062 y Ft(\()p Fn(a)467 5076 y Fq(2)507 5062 y Fn(;)p 547 5012 V 15 w(a)595 5076 y Fq(0)635 5062 y Ft(\))g(=)g(2)f(hence)g(that)h(an)m(y)f(solution)f(of)h(\(1\))h(has)e (a)i(denominator)e(of)h(the)g(form)f Fn(x)3533 5029 y Fh(n)p 3580 4988 48 4 v 3580 5062 a Fn(d)h Ft(where)0 5174 y(Dis)131 5188 y Fh(\033)177 5174 y Ft(\()p 212 5100 V Fn(d)q Ft(\))26 b Fi(\024)f Ft(0.)42 b(Using)30 b(the)h(b)s(ound)d Fn(h)e Ft(=)f(1,)32 b(w)m(e)f(get)g Fn(L)1845 5189 y Fh(h)1916 5174 y Ft(=)25 b Fn(L)30 b Ft(and)p 2281 5100 V 30 w Fn(d)h Ft(divides)d(the)j(greatest)i(common)d (divisor)f(of)0 5283 y(gcd)136 5304 y Fq(0)p Fg(\024)p Fh(i)p Fg(\024)p Fq(2)345 5283 y Ft(\()p Fn(\033)435 5250 y Fg(\000)p Fh(i)519 5283 y Fn(a)567 5297 y Fh(i)595 5283 y Ft(\))c(=)g(gcd)888 5209 y Fe(\000)929 5283 y Fn(x)t Ft(+)t Fn(q)s(;)15 b(\033)1199 5250 y Fg(\000)p Fq(1)1293 5283 y Ft(\()p Fn(q)1372 5250 y Fq(2)1412 5283 y Ft(\()p Fn(x)t Ft(+)t(1\)\))p Fn(;)g(\033)1788 5250 y Fg(\000)p Fq(2)1883 5283 y Ft(\()p Fn(q)1962 5250 y Fq(3)2002 5283 y Ft(\()p Fn(q)s(x)t Ft(+)t(1\)\))2327 5209 y Fe(\001)2394 5283 y Ft(=)25 b(gcd)2626 5209 y Fe(\000)2668 5283 y Fn(x)t Ft(+)t Fn(q)s(;)15 b(q)s Ft(\()p Fn(x)t Ft(+)t Fn(q)s Ft(\))p Fn(;)g(q)3256 5250 y Fq(2)3294 5283 y Ft(\()p Fn(x)t Ft(+)t Fn(q)s Ft(\))3539 5209 y Fe(\001)3605 5283 y Ft(=)25 b Fn(x)t Ft(+)t Fn(q)s Ft(.)p eop %%Page: 69 5 69 4 bop 2541 66 a Ff(M.)23 b(Bronstein,)g(summary)e(b)n(y)j(A.)f(F)-6 b(redet)142 b Fq(69)0 266 y Ft(Therefore,)34 b(an)m(y)f(rational)f (solution)f(of)j(\(1\))g(can)f(b)s(e)f(written)g(as)h Fn(y)g Ft(=)c Fn(p=)2557 192 y Fe(\000)2599 266 y Fn(x)2651 233 y Fh(n)2697 266 y Ft(\()p Fn(x)23 b Ft(+)e Fn(q)s Ft(\))2978 192 y Fe(\001)3053 266 y Ft(where)32 b Fn(n)d Fi(\025)h Ft(0)j(and)f Fn(p)h Ft(is)0 374 y(in)c Fl(Q)9 b Ft([)p Fn(x)p Ft(].)100 482 y(The)21 b(indicial)e(equation)i(at)i Fn(x)i Ft(=)g(0)d(is)f Fn(q)s(Z)1489 449 y Fq(2)1531 482 y Fi(\000)s Ft(2)p Fn(q)1694 449 y Fq(2)1733 482 y Fn(Z)10 b Ft(+)s Fn(q)1923 449 y Fq(3)1987 482 y Ft(=)25 b(0)d(\(see)h([2)q(]\).)38 b(Its)22 b(only)f(solution)f(of)i(the)g (form)f Fn(Z)32 b Ft(=)25 b Fn(q)3853 449 y Fh(n)0 590 y Ft(is)34 b(for)g Fn(n)f Ft(=)f(1,)37 b(whic)m(h)c(implies)f(that)k (an)m(y)f(rational)f(solution)g(of)h(\(1\))g(can)h(b)s(e)e(written)g (as)h Fn(y)g Ft(=)e Fn(p=)3456 517 y Fe(\000)3498 590 y Fn(x)p Ft(\()p Fn(x)23 b Ft(+)g Fn(q)s Ft(\))3833 517 y Fe(\001)3875 590 y Ft(.)0 706 y(Replacing)31 b Fn(y)j Ft(b)m(y)d(this)g(form,)g(w)m(e)h(get)h Fn(p)p Ft(\()p Fn(q)1461 673 y Fq(2)1501 706 y Fn(x)p Ft(\))21 b Fi(\000)g Ft(2)p Fn(p)p Ft(\()p Fn(q)s(x)p Ft(\))g(+)g Fn(p)p Ft(\()p Fn(x)p Ft(\))28 b(=)e(0)32 b(\(whose)g(solution)e(space)i(is)f Fl(Q)9 b Ft(\()p Fn(q)s Ft(\),)38 b(whic)m(h)0 814 y(implies)28 b(that)j(the)f(general)h(rational)e(solution)g(of)i(\(1\))g(is)e Fn(y)g Ft(=)c Fn(C)t(=)2283 741 y Fe(\000)2325 814 y Fn(x)p Ft(\()p Fn(x)20 b Ft(+)g Fn(q)s Ft(\))2654 741 y Fe(\001)2726 814 y Ft(for)30 b(an)m(y)h Fn(C)37 b Ft(in)29 b Fl(Q)9 b Ft(\()p Fn(q)s Ft(\)\).)0 984 y(5.2.)47 b Fs(Lauren)m(t)40 b(p)s(olynomial)f(solution.)47 b Ft(The)34 b(problem)f(of)i(\014nding)e(rational)h(solutions)g Fn(y)j Ft(of)e Fn(L)p Ft(\()p Fn(y)s Ft(\))f(=)e Fn(b)j Ft(is)0 1102 y(reduced)d(to)h(\014nding)e Fn(y)k Ft(in)d Fn(k)s Ft([)p Fn(t;)15 b(t)1130 1069 y Fg(\000)p Fq(1)1225 1102 y Ft(])32 b(suc)m(h)h(that)g Fn(L)p Ft(\()p Fn(y)s Ft(\))d(=)f Fn(b)p Ft(,)k(where)f Fn(b)h Ft(is)f(in)f Fn(k)s Ft([)p Fn(t;)15 b(t)2816 1069 y Fg(\000)p Fq(1)2911 1102 y Ft(])33 b(and)f Fn(L)d Ft(=)3339 1034 y Fe(P)3435 1060 y Fh(N)3435 1129 y(i)p Fq(=0)3569 1102 y Fn(a)3617 1116 y Fh(i)3645 1102 y Fn(\033)3700 1069 y Fh(i)3761 1102 y Ft(is)j(a)0 1210 y(di\013erence)e(op)s(erator,)h(with)e Fn(a)1054 1224 y Fh(i)1107 1210 y Fi(2)c Fn(k)s Ft([)p Fn(t)p Ft(])31 b(and)f(non-zero)h Fn(a)1951 1224 y Fq(0)2020 1210 y Ft(and)f Fn(a)2245 1224 y Fh(N)2313 1210 y Ft(.)40 b(This)29 b(decomp)s(oses)h(in)f(t)m(w)m(o)j(steps:)145 1339 y(1.)42 b(\014nd)29 b(a)i(b)s(ound)d(for)i(the)h(degree)g(and)f(the)g(order)g (in)f Fn(t)h Ft(of)h Fn(y)s Ft(;)145 1447 y(2.)42 b(compute)31 b(the)g(co)s(e\016cien)m(ts)g(of)f Fn(y)s Ft(,)h(seen)f(as)h(a)f (Lauren)m(t)h(p)s(olynomial)d(in)h Fn(t)p Ft(.)0 1622 y(5.2.1.)48 b Fo(Bound)28 b(for)g(the)h(de)-5 b(gr)g(e)g(e)28 b(and)h(or)-5 b(der)29 b(of)f(a)h(p)-5 b(olynomial)30 b(solution.)47 b Ft(One)25 b(rewrites)f Fn(L)h Ft(as)3233 1554 y Fe(P)3329 1580 y Fh(d)3329 1649 y(j)t Fq(=)p Fh(\027)3475 1622 y Fn(t)3508 1589 y Fh(j)3544 1622 y Fn(L)3606 1636 y Fh(j)3667 1622 y Ft(where)0 1749 y(the)33 b Fn(L)221 1763 y Fh(j)257 1749 y Ft('s)g(are)g(in)e Fn(k)s Ft([)p Fn(\033)s Ft(])i(and)f Fn(L)1042 1763 y Fh(\027)1117 1749 y Ft(and)g Fn(L)1358 1764 y Fh(d)1431 1749 y Ft(are)h(not)g(equal) f(to)h(zero.)48 b(Let)33 b Fn(y)f Ft(=)c Fn(y)2724 1764 y Fh(\016)2762 1749 y Fn(t)2795 1716 y Fh(\016)2854 1749 y Ft(+)21 b Fi(\001)15 b(\001)g(\001)23 b Ft(+)e Fn(y)3211 1763 y Fh(\015)3255 1749 y Fn(t)3288 1716 y Fh(\015)3365 1749 y Ft(b)s(e)32 b(in)f Fn(k)s Ft([)p Fn(t;)15 b(t)3780 1716 y Fg(\000)p Fq(1)3875 1749 y Ft(])0 1857 y(for)32 b(in)m(tegers)h Fn(\015)38 b Ft(and)31 b Fn(\016)37 b Ft(satisfying)31 b Fn(\015)j Fi(\025)28 b Fn(\016)36 b Ft(and)c(suc)m(h)h(that)g(neither)e Fn(y)2424 1872 y Fh(\016)2494 1857 y Ft(nor)h Fn(y)2703 1871 y Fh(\015)2779 1857 y Ft(is)g(equal)g(to)h(zero.)48 b(Let)33 b Fn(b)f Ft(b)s(e)g(in)0 1968 y Fn(k)s Ft([)p Fn(t;)15 b(t)181 1935 y Fg(\000)p Fq(1)276 1968 y Ft(].)41 b(If)30 b Fn(L)p Ft(\()p Fn(y)s Ft(\))25 b(=)g Fn(b)p Ft(,)31 b(then)145 2099 y(1.)42 b(either)30 b Fn(\016)f Fi(\025)c Fn(\027)6 b Ft(\()p Fn(b)p Ft(\))21 b Fi(\000)f Fn(\027)6 b Ft(,)30 b(or)g Fn(L)1231 2113 y Fh(\027)1274 2099 y Ft(\()p Fn(y)1354 2114 y Fh(\016)1392 2099 y Fn(t)1425 2066 y Fh(\016)1463 2099 y Ft(\))25 b(=)g(0;)145 2207 y(2.)42 b(either)30 b Fn(\015)g Fi(\024)25 b Ft(deg)17 b Fn(b)j Fi(\000)g Fn(d)p Ft(,)31 b(or)g Fn(L)1268 2222 y Fh(d)1308 2207 y Ft(\()p Fn(y)1388 2221 y Fh(\015)1432 2207 y Fn(t)1465 2174 y Fh(\015)1510 2207 y Ft(\))25 b(=)g(0.)0 2350 y(The)38 b(problem)g(is)f(reduced)i(to)g(considering)e(di\013erence)h(op)s (erators)h Fn(T)52 b Ft(=)2650 2281 y Fe(P)2745 2308 y Fh(M)2745 2376 y(i)p Fq(=)p Fh(m)2906 2350 y Fn(A)2974 2364 y Fh(i)3002 2350 y Fn(\033)3057 2317 y Fh(i)3125 2350 y Ft(with)37 b Fn(A)3408 2364 y Fh(i)3476 2350 y Fi(2)i Fn(C)7 b Ft([)p Fn(n)p Ft(])38 b(for)0 2457 y(non-zero)29 b Fn(A)435 2471 y Fh(m)529 2457 y Ft(and)f Fn(A)772 2471 y Fh(M)851 2457 y Ft(,)h(and)e(to)i(searc)m(hing)f(b)s(ounds)e(for)i Fn(\015)i Fi(2)25 b Fl(Z)f Ft(suc)m(h)j(that)i Fn(T)13 b Ft(\()p Fn(z)t(t)2863 2425 y Fh(\015)2908 2457 y Ft(\))25 b(=)g(0)k(for)e(some)i Fn(z)j Ft(in)27 b Fn(C)7 b Ft(\()p Fn(n)p Ft(\).)0 2565 y(Let)31 b Fn(e)25 b Ft(=)g Fi(\000)p Fn(\027)442 2579 y Fg(1)517 2565 y Ft(\()p Fn(\033)s(t=t)p Ft(\))h(=)f Fn(\027)920 2579 y Fg(1)995 2565 y Ft(\()p Fn(a)p Ft(\).)41 b(There)30 b(are)h(three)f(p)s(ossibilities:)164 2694 y Fs({)41 b Ft(if)30 b Fn(e)25 b(>)g Ft(0)31 b(then)f(\(deg)960 2716 y Fh(n)1022 2694 y Fn(A)1090 2708 y Fh(m)1177 2694 y Fi(\000)20 b Ft(deg)1406 2716 y Fh(n)1468 2694 y Fn(T)13 b Ft(\))p Fn(=e)26 b Fi(\024)f Fn(\015)30 b Fi(\024)25 b Ft(\(deg)2124 2716 y Fh(n)2186 2694 y Fn(T)33 b Fi(\000)20 b Ft(deg)2501 2716 y Fh(n)2563 2694 y Fn(A)2631 2708 y Fh(M)2710 2694 y Ft(\))p Fn(=e)p Ft(;)164 2802 y Fs({)41 b Ft(if)30 b Fn(e)25 b(<)g Ft(0)31 b(then)f(\(deg)960 2824 y Fh(n)1022 2802 y Fn(T)j Fi(\000)20 b Ft(deg)1337 2824 y Fh(n)1399 2802 y Fn(A)1467 2816 y Fh(m)1533 2802 y Ft(\))p Fn(=e)27 b Fi(\024)e Fn(\015)30 b Fi(\024)25 b Ft(\(deg)2124 2824 y Fh(n)2186 2802 y Fn(A)2254 2816 y Fh(m)2341 2802 y Fi(\000)20 b Ft(deg)2570 2824 y Fh(n)2632 2802 y Fn(T)13 b Ft(\))p Fn(=e)p Ft(;)164 2910 y Fs({)41 b Ft(if)f Fn(e)j Ft(=)f(0)f(then)g Fn(\013)i Ft(=)f Fn(a)p Ft(\()p Fi(1)p Ft(\))h Fi(2)f Fn(C)1494 2877 y Fh(?)1533 2910 y Ft(.)72 b(W)-8 b(e)42 b(decomp)s(ose)f Fn(A)2333 2924 y Fh(i)2404 2910 y Ft(=)h Fn(a)2565 2924 y Fh(i;\013)2654 2934 y Fc(i)2685 2910 y Fn(n)2740 2877 y Fh(\013)2785 2887 y Fc(i)2842 2910 y Ft(+)27 b Fi(\001)15 b(\001)g(\001)h Ft(.)72 b(W)-8 b(e)42 b(de\014ne)e Fn(Q)p Ft(\()p Fn(z)t Ft(\))j(=)257 2956 y Fe(P)353 3051 y Fh(i)p Fg(j)p Fh(\013)442 3061 y Fc(i)469 3051 y Fq(=max)655 3061 y Fc(j)687 3051 y Fq(\()p Fh(\013)759 3061 y Fc(j)793 3051 y Fq(\))839 3024 y Fn(a)887 3038 y Fh(i;\013)976 3048 y Fc(i)1007 3024 y Fn(z)1053 2991 y Fh(i)1081 3024 y Ft(.)63 b(W)-8 b(e)39 b(ha)m(v)m(e)h Fn(Q)p Ft(\()p Fn(\013)1716 2991 y Fh(\015)1761 3024 y Ft(\))e(=)f(0.)64 b(This)36 b(problem)g(can)i(b)s 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Fn(z)t Ft(\))c(=)f Fn(b)32 b Ft(implies)d Fn(L)3278 3932 y Fh(j)3314 3918 y Ft(\()p Fn(z)t Ft(\))g(=)e Fn(b)3596 3932 y Fh(j)3664 3918 y Ft(for)k(all)257 4034 y Fn(j)36 b Ft(and)30 b(this)f(reduces)h(to)h(di\013erence)f(equations)g(with)f (co)s(e\016cien)m(ts)i(in)f Fn(C)7 b Ft([)p Fn(n)p Ft(];)164 4142 y Fs({)41 b Ft(if)28 b Fn(J)35 b(>)25 b Ft(0)k(then)g(one)h (decomp)s(oses)f Fn(z)g Ft(=)c Fn(z)1667 4156 y Fq(0)1725 4142 y Ft(+)17 b Fn(t)p 1846 4092 47 4 v(z)34 b Ft(where)28 b Fn(z)2225 4156 y Fq(0)2290 4142 y Ft(=)d Fn(z)t Ft(\(0\))31 b(is)d(in)g Fn(C)7 b Ft(\()p Fn(n)p Ft(\).)40 b(Then)28 b Fn(L)3333 4156 y Fq(0)3372 4142 y Ft(\()p Fn(z)3449 4156 y Fq(0)3489 4142 y Ft(\))e(=)f Fn(b)3685 4156 y Fq(0)3754 4142 y Ft(and)257 4250 y(one)31 b(can)g(\014nd)d Fn(z)814 4264 y Fq(0)854 4250 y Ft(.)41 b(So,)30 b Fn(L)p Ft(\()p Fn(z)t Ft(\))c(=)f(\()p Fn(L)c Fi(\000)f Fn(L)1642 4264 y Fq(0)1681 4250 y Ft(\)\()p Fn(z)1793 4264 y Fq(0)1833 4250 y Ft(\))h(+)f Fn(L)p Ft(\()p Fn(t)p 2110 4200 V(z)t Ft(\))h(+)f Fn(L)2365 4264 y Fq(0)2404 4250 y Ft(\()p Fn(z)2481 4264 y Fq(0)2521 4250 y Ft(\))31 b(and)f Fn(L)p Ft(\()p Fn(z)t Ft(\))c(=)f Fn(b)30 b Ft(implies)1215 4400 y Fn(L)p Ft(\()p Fn(t)p 1345 4350 V(z)5 b Ft(\))83 b(=)g Fn(b)20 b Fi(\000)g Fn(b)1853 4414 y Fq(0)1913 4400 y Fi(\000)f Ft(\()p Fn(L)i Fi(\000)f Fn(L)2274 4414 y Fq(0)2313 4400 y Ft(\))p Fn(z)2390 4414 y Fq(0)1215 4590 y Fn(t)1259 4567 y Ft(~)1248 4590 y Fn(L)p Ft(\()p 1345 4539 V Fn(z)5 b Ft(\))83 b(=)g Fn(t)1712 4461 y Fe(\022)1789 4528 y Fn(b)20 b Fi(\000)g Fn(b)1978 4542 y Fq(0)p 1789 4569 229 4 v 1887 4652 a Fn(t)2027 4461 y Fe(\023)2114 4590 y Fi(\000)g Fn(t)2248 4528 y Ft(\()p Fn(L)g Fi(\000)g Fn(L)2518 4542 y Fq(0)2558 4528 y Ft(\))p Fn(z)2635 4542 y Fq(0)p 2248 4569 427 4 v 2445 4652 a Fn(t)257 4789 y Ft(This)31 b(giv)m(es)h(us)g(a)h(new)f(di\013erence)f (equation)h(with)f(a)i(solution)p 2491 4739 47 4 v 31 w Fn(z)k Ft(of)32 b(degree)h(strictly)e(less)h(than)g Fn(J)9 b Ft(.)47 b(By)257 4897 y(induction,)29 b(one)h(can)h(\014nd)p 1202 4847 V 29 w Fn(z)t Ft(.)0 5025 y Fo(Example.)47 b Ft(Consider)36 b Fn(y)s Ft(\()p Fn(n)25 b Ft(+)g(2\))h Fi(\000)f Ft(\()p Fn(n)p Ft(!)h(+)f Fn(n)p Ft(\))p Fn(y)s Ft(\()p Fn(n)f Ft(+)h(1\))i(+)e Fn(n)p Ft(\()p Fn(n)p Ft(!)g Fi(\000)g Ft(1\))p Fn(y)s Ft(\()p Fn(n)p Ft(\))39 b(=)e(0,)k(whic)m(h)36 b(is)i(asso)s(ciated)g(to)h(the)0 5133 y(di\013erence)30 b(op)s(erator)509 5283 y Fn(L)25 b Ft(=)g Fn(\033)747 5245 y Fq(2)807 5283 y Fi(\000)20 b Ft(\()p Fn(t)g Ft(+)g Fn(n)p Ft(\))p Fn(\033)k Ft(+)c Fn(n)p Ft(\()p Fn(t)g Fi(\000)f Ft(1\))27 b(=)d Fn(t)p Ft(\()p Fn(n)c Fi(\000)g Fn(\033)s Ft(\))h(+)f(\()p Fn(\033)2295 5245 y Fq(2)2355 5283 y Fi(\000)g Fn(n\033)j Fi(\000)d Fn(n)p Ft(\))25 b(=)g Fn(tL)2973 5297 y Fq(1)3032 5283 y Ft(+)20 b Fn(L)3185 5297 y Fq(0)p eop %%Page: 70 6 70 5 bop 0 66 a Fq(70)142 b Ff(Di\013erence)24 b(Equations)h(with)e (Hyp)r(ergeometric)h(Co)r(e\016cien)n(ts)0 266 y Ft(Using)31 b(the)g(same)h(notations)f(as)h(previously)-8 b(,)30 b Fn(e)d Ft(=)g Fi(\000)p Fn(\027)1893 280 y Fg(1)1967 266 y Ft(\()p Fn(\033)s(t=t)p Ft(\))h(=)f Fi(\000)p Fn(\027)2445 280 y Fg(1)2519 266 y Ft(\()p Fn(n)21 b Ft(+)f(1\))28 b(=)f(1)k(and)g(then)g Fn(y)f Ft(=)c Fn(y)3606 280 y Fq(0)3666 266 y Ft(+)21 b Fn(y)3803 280 y Fq(1)3842 266 y Fn(t)p Ft(.)0 374 y(One)30 b(\014rst)g(considers)f Fn(L)832 388 y Fq(0)871 374 y Ft(\()p Fn(y)951 388 y Fq(0)990 374 y Ft(\))d(=)f Fn(\033)1202 341 y Fq(2)1242 374 y Fn(y)1287 388 y Fq(0)1346 374 y Fi(\000)20 b Fn(n\033)s(y)1592 388 y Fq(0)1651 374 y Fi(\000)g Fn(ny)1842 388 y Fq(0)1905 374 y Ft(=)25 b(0,)31 b(and)f(\014nds)f(that)i Fn(y)2739 388 y Fq(0)2803 374 y Ft(=)25 b(0.)41 b(Then:)608 523 y Fn(L)p Ft(\()p Fn(tz)780 537 y Fq(1)819 523 y Ft(\))26 b(=)f(\()p Fn(n)20 b Ft(+)g(2\)\()p Fn(n)h Ft(+)f(1\))p Fn(t\033)1627 485 y Fq(2)1667 523 y Ft(\()p Fn(z)1744 537 y Fq(1)1784 523 y Ft(\))h Fi(\000)f Ft(\()p Fn(n)g Ft(+)g(1\)\()p Fn(t)h Ft(+)f Fn(n)p Ft(\))p Fn(t\033)s Ft(\()p Fn(y)2650 537 y Fq(1)2689 523 y Ft(\))h(+)f Fn(n)p Ft(\()p Fn(t)g Fi(\000)f Ft(1\))p Fn(ty)3227 537 y Fq(1)3267 523 y Fn(;)0 668 y Ft(from)30 b(whic)m(h)f(follo)m(ws)g(that)598 791 y(~)588 814 y Fn(L)p Ft(\()p Fn(y)730 828 y Fq(1)769 814 y Ft(\))d(=)e(\()p Fn(n)d Ft(+)f(2\)\()p Fn(n)g Ft(+)g(1\))p Fn(\033)1543 777 y Fq(2)1584 814 y Ft(\()p Fn(y)1664 828 y Fq(1)1703 814 y Ft(\))h Fi(\000)f Ft(\()p Fn(n)g Ft(+)g(1\)\()p Fn(t)h Ft(+)f Fn(n)p Ft(\))p Fn(\033)s Ft(\()p Fn(y)2536 828 y Fq(1)2575 814 y Ft(\))h(+)f Fn(n)p Ft(\()p Fn(t)g Fi(\000)g Ft(1\))p Fn(y)3081 828 y Fq(1)3146 814 y Ft(=)25 b(0)p Fn(:)0 960 y Ft(This)k(implies)e(that)k Fn(y)758 974 y Fq(1)822 960 y Ft(=)25 b Fn(c=n)p Ft(.)41 b(Then)30 b Fn(y)e Ft(=)d Fn(y)1575 974 y Fq(1)1614 960 y Fn(t)g Ft(=)g(\()p Fn(c=n)p Ft(\))p Fn(n)p Ft(!)h(=)f Fn(c)p Ft(\()p Fn(n)20 b Fi(\000)g Ft(1\)!.)1682 1135 y Fp(Bibliograph)m(y)38 1280 y Fv([1])40 b(Abramo)n(v)21 b(\(S.)h(A.\).)h({)g(Rational)g(solutions)h(of)f(linear)g(di\013eren)n 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b(Computation)p Fv(.)h({)e(T)-6 b(o)26 b(app)r(ear.)g(Preliminary)g(v)n(ersion)g(a)n(v)l(ailable)g(as)h (INRIA)c(Researc)n(h)j(Rep)r(ort)f(n)3362 2382 y(\027)3420 2376 y(3797.)38 2468 y([6])40 b(Bronstein)28 b(\(Man)n(uel\).)f({)h(On) e(solutions)j(of)f(linear)g(ordinary)f(di\013eren)n(tial)h(equations)f (in)g(their)g(co)r(e\016cien)n(t)h(\014eld.)g Fr(Journal)h(of)158 2559 y(Symb)l(olic)e(Computation)p Fv(,)h(v)n(ol.)e(13,)g(n)1250 2565 y(\027)1307 2559 y(4,)h Fd(1992)q Fv(,)f(pp.)f(413{439.)38 2650 y([7])40 b(Bronstein)25 b(\(Man)n(uel\))f(and)f(F)-6 b(redet)24 b(\(Anne\).)f({)i(Solving)f(linear)h(ordinary)f(di\013eren)n (tial)g(equations)h(o)n(v)n(er)e Fd(C)3304 2590 y Fa(\000)3340 2650 y Fd(x;)12 b Fv(exp)3547 2590 y Fa(R)3611 2650 y Fd(f)c Fv(\()p Fd(x)p Fv(\))p Fd(dx)3843 2590 y Fa(\001)3879 2650 y Fv(.)158 2742 y(In)28 b(Do)r(oley)i(\(Sam\))e(\(editor\),)i Fr(ISSA)n(C'99)g(\(July)h(29{31,)h(1999\))p Fv(.)f(pp.)d(173{179.)k({)e (A)n(CM)f(Press,)i Fd(1999)r 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Fr(On)g(the)h(inte)l(gr)l(ation)h(of)d(elementary) j(functions)f(which)f(ar)l(e)h(built)f(up)h(using)f(algebr)l(aic)h(op)l (er)l(ations)p Fv(.)h({)158 3472 y(Rep)r(ort)25 b(n)441 3478 y(\027)498 3472 y(SP-2801/002/00,)30 b(Sys.)c(Dev.)f(Corp.,)i(San) n(ta)e(Monica,)i(CA,)f Fd(1968)r Fv(.)0 3563 y([12])40 b(Risc)n(h)d(\(Rob)r(ert)g(H.\).)g({)h(The)f(problem)g(of)h(in)n (tegration)g(in)f(\014nite)g(terms.)g Fr(T)-6 b(r)l(ansactions)40 b(of)e(the)h(AMS)p Fv(,)f(v)n(ol.)f(139,)42 b Fd(1969)r Fv(,)158 3655 y(pp.)25 b(167{189.)0 3746 y([13])40 b(Risc)n(h)30 b(\(Rob)r(ert)g(H.\).)g({)h(The)g(solution)g(of)g(the)f(problem)g(of)h (in)n(tegration)g(in)g(\014nite)f(terms.)g Fr(Bul)t(letin)h(of)h(the)h (AMS)p Fv(,)d(v)n(ol.)h(76,)158 3837 y Fd(1970)q Fv(,)26 b(pp.)g(605{608.)0 3929 y([14])40 b(Singer)28 b(\(Mic)n(hael)g(F.\).)g ({)g(Liouvillian)h(solutions)f(of)h(linear)f(di\013eren)n(tial)g (equations)g(with)f(Liouvillian)i(co)r(e\016cien)n(ts.)g Fr(Journal)158 4020 y(of)e(Symb)l(olic)h(Computation)p Fv(,)f(v)n(ol.)f(11,)g(n)1340 4026 y(\027)1398 4020 y(3,)g Fd(1991)q Fv(,)g(pp.)g(251{273.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF/tvz@princeton.edu