{VERSION 7 1 "Linux" "7.1" } {USTYLETAB {PSTYLE "Ordered List 5" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 } {PSTYLE "Heading 4" -1 20 1 {CSTYLE "" -1 -1 "MS Serif" 1 12 0 0 0 1 1 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output " -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 3 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "MS S erif" 1 14 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 12 40 120 40 1 2 2 2 2 2 1 2 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered \+ List 4" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List Item" -1 14 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 } {PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 2 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "H eading 2" -1 4 1 {CSTYLE "" -1 -1 "MS Serif" 1 16 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courie r" 1 12 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "MS Serif" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered Li st 2" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "Equation Label" -1 200 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 201 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }{CSTYLE "Dictionary Hyperlink" -1 45 "MS Serif" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 0 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 202 "Couri er" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "MS Ser if" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 203 29 "Komputeralgebrai algorit musok" }}}{EXCHG {PARA 19 "" 0 "" {TEXT 204 18 "J\303\241rai Antal" }} }{EXCHG {PARA 19 "" 0 "" {TEXT 201 69 "Ezek a programok csak szeml\303 \251ltet\303\251sre szolg\303\241lnak." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 4 "1. T" }{TEXT 205 18 "\303\266rt\303\251" }{TEXT 205 3 "ne t" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 18 "2. Algebrai alapok" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT 205 7 "3. Norm" }{TEXT 205 8 "\303\241" }{TEXT 205 6 "l form" }{TEXT 205 8 "\303\241" }{TEXT 205 12 "k, reprez ent" }{TEXT 205 18 "\303\241ci\303\263" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 13 "4. Aritmetika" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 4 "5 . K" }{TEXT 205 8 "\303\255" }{TEXT 205 9 "nai marad" }{TEXT 205 8 "\3 03\251" }{TEXT 205 3 "kol" }{TEXT 205 8 "\303\241" }{TEXT 205 1 "s" }} }{SECT 0 {PARA 3 "" 0 "" {TEXT 205 14 "6. Newton-iter" }{TEXT 205 8 " \303\241" }{TEXT 205 2 "ci" }{TEXT 205 8 "\303\263" }{TEXT 205 16 ", H ensel-felemel" }{TEXT 205 8 "\303\251" }{TEXT 205 1 "s" }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart; " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.1. P" }{TEXT 206 12 "\30 3\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "`mod`:=mods; p:=97; u:=-272300; u0:=u mod p; u1:=(u- u0)/p mod p;\n" }{MPLTEXT 1 0 48 "u2:=(u-(u0+u1*p))/p^2 mod p; u-(u0+u 1*p+u2*p^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%modsG%*protectedG" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"#(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "! '+BF" }}{PARA 11 "" 1 "" {XPPMATH 20 "!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "% #%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.2. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "p:=5; u:=14*x^2-11*x-15; u0:=u mod p; u1:=(u- u0)/p mod p;\n" }{MPLTEXT 1 0 48 "u2:=(u-(u0+u1*p))/p^2 mod p; u-(u0+u 1*p+u2*p^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 " " {XPPMATH 20 ",(*&\"#9\"\"\")I\"xG6\"\"\"#F%F%*&\"#6F%F'F%!\"\"\"#:F, " }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"!\"\"F%F)" } }{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"#\"\"\")I\"xG6\"F$F%!\"\"*&F$F% F'F%F)F$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F( F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.3. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "u:='u'; a:=36*x^4-180*x^3 +93*x^2+330*x+121;\n" }{MPLTEXT 1 0 24 "F:=a-u^2; Fp:=diff(F,u);" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"uG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&\"#O\"\"\")I\"xG6\"\"\"%F%F%*&\"$!=F%)F'\"\"$F%!\"\"*&\"#$*F%)F' \"\"#F%F%*&\"$I$F%F'F%F%\"$@\"F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*& \"#O\"\"\")I\"xG6\"\"\"%F%F%*&\"$!=F%)F'\"\"$F%!\"\"*&\"#$*F%)F'\"\"#F %F%*&\"$I$F%F'F%F%\"$@\"F%*$)I\"uGF(F2F%F." }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"uG6\"F%!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"x G6\"\"\"%\"\"\"F(*&\"\"#F()F%F*F(!\"\"F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 40 "u0:=x^2-1; u[1]:=u0; d:=subs(u=u[1],Fp);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%!\"\"F$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "u1:=-expand(subs(u=u[1],F)/p); u1:= Quo(u1,d,x) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&\"\"(\"\"\")I \"xG6\"\"\"%F%!\"\"*&\"#OF%)F'\"\"$F%F%*&\"#>F%)F'\"\"#F%F**&\"#mF%F'F %F*\"#CF*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(* &F'F(F%F(F(F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "u[2]: =u0+5*u1; u2:=-expand(subs(u=u[2],F)/p^2); u2:=Quo(u2,d,x) mod p;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"'\"\"\")I\"xG6\"\"\"#F%F%\"#6!\" \"*&\"#5F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"#7\"\"\")I\"xG 6\"\"\"$F%F%*&\"\"&F%)F'\"\"#F%!\"\"*&\"#AF%F'F%F." }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"xG6\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "u[3]:=u0+5*u1+5^2*u2; expand(subs(u=u[3],F));" }}{PARA 11 "" 1 " " {XPPMATH 20 ",(*&\"\"'\"\"\")I\"xG6\"\"\"#F%F%\"#6!\"\"*&\"#:F%F'F%F +" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.4. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p:=5; `mod` :=mods;\n" }{MPLTEXT 1 0 105 "a:=x^4+x^3*y^2-x^2*y^4+x^2*y*z+2*x^2*z-2 *x^2-2*x*y^3*z+x*y^2*z-x*y^2-y^2*z^2+y*z^2-y*z+z^2-2*z+1 mod p;\n" } {MPLTEXT 1 0 56 "a:=collect(a,[y,z],`distributed`); sort(a,[y,z],tdeg) ;\n" }{MPLTEXT 1 0 24 "F:=a-u^2; Fp:=diff(F,u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "I%modsG%*protected G" }}{PARA 11 "" 1 "" {XPPMATH 20 ",@*$)I\"xG6\"\"\"%\"\"\"F(*&)F%\"\" $F()I\"yGF&\"\"#F(F(*&)F%F.F()F-F'F(!\"\"*(F0F(F-F(I\"zGF&F(F(*(F.F(F0 F(F4F(F(*&F.F(F0F(F2**F.F(F%F()F-F+F(F4F(F2*(F%F(F,F(F4F(F(*&F%F(F,F(F 2*&F,F()F4F.F(F2*&F-F(F " 0 "" {MPLTEXT 1 0 29 "subs(y=0,z=0,a); u[1]:=x^2-1;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"%\"\"\"F(*&\"\"#F()F%F*F(!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F(!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 25 "d:=subs(u=u[1],Fp) mod p;" }}{PARA 11 "" 1 " " {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%!\"\"F$F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "FF:=expand(subs(u=u[1],F)) mod p;\n" } {MPLTEXT 1 0 37 "FF:=collect(%,[y,z],`distributed`);\n" }{MPLTEXT 1 0 21 "sort(%,[y,z],tdeg);\n" }{MPLTEXT 1 0 37 "u2:=0; u3:=-Quo(2*x^2-2,d ,x) mod p;\n" }{MPLTEXT 1 0 35 "du[2]:=u2*y+u3*z; u[2]:=u[1]+du[2];" } }{PARA 11 "" 1 "" {XPPMATH 20 ",:*&)I\"xG6\"\"\"#\"\"\")I\"yGF&\"\"%F( !\"\"**F'F(F%F()F*\"\"$F(I\"zGF&F(F,*&)F*F'F()F0F'F(F,*(F%F(F2F(F0F(F( *&F*F(F3F(F(*&F%F(F2F(F,*&)F%F/F(F2F(F(*(F$F(F*F(F0F(F(*&F*F(F0F(F,*$F 3F(F(*(F'F(F$F(F0F(F(*&F'F(F0F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",4* *\"\"#\"\"\"I\"xG6\"F%)I\"yGF'\"\"$F%I\"zGF'F%!\"\"*(F&F%)F)F$F%F+F%F% *&F.F%)F+F$F%F,*&F)F%F0F%F%*(,&*$)F&F$F%F%F%F,F%F)F%F+F%F%*&,&*&F$F%F5 F%F%F$F,F%F+F%F%*&,&F&F,*$)F&F*F%F%F%F.F%F%*$F0F%F%*&F5F%)F)\"\"%F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",4*&)I\"xG6\"\"\"#\"\"\")I\"yGF&\"\"% F(!\"\"**F'F(F%F()F*\"\"$F(I\"zGF&F(F,*&)F*F'F()F0F'F(F,*(F%F(F2F(F0F( F(*&F*F(F3F(F(*&,&F%F,*$)F%F/F(F(F(F2F(F(*(,&*$F$F(F(F(F,F(F*F(F0F(F(* $F3F(F(*&,&*&F'F(F$F(F(F'F,F(F0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\" \"#\"\"\"F(F(!\"\"I\"zGF&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "FF:=expand(subs(u=u[2],F)) mod p;\n" }{MPLTEXT 1 0 37 "FF:=collect (%,[y,z],`distributed`);\n" }{MPLTEXT 1 0 21 "sort(%,[y,z],tdeg);\n" } {MPLTEXT 1 0 81 "u22:=-Quo(x^3-x,d,x) mod p; u23:=-Quo(x^2-1,d,x) mod \+ p; u33:=-Quo(0,d,x) mod p;\n" }{MPLTEXT 1 0 49 "du[3]:=u22*y^2+u23*y*z +u33*z^2; u[3]:=u[2]+du[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 ",4*&)I\"x G6\"\"\"#\"\"\")I\"yGF&\"\"%F(!\"\"**F'F(F%F()F*\"\"$F(I\"zGF&F(F,*&)F *F'F()F0F'F(F,*(F%F(F2F(F0F(F(*&F*F(F3F(F(*&F%F(F2F(F,*&)F%F/F(F2F(F(* (F$F(F*F(F0F(F(*&F*F(F0F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",0**\"\"# \"\"\"I\"xG6\"F%)I\"yGF'\"\"$F%I\"zGF'F%!\"\"*(F&F%)F)F$F%F+F%F%*&F.F% )F+F$F%F,*&F)F%F0F%F%*(,&*$)F&F$F%F%F%F,F%F)F%F+F%F%*&,&F&F,*$)F&F*F%F %F%F.F%F%*&F5F%)F)\"\"%F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&)I\"x G6\"\"\"#\"\"\")I\"yGF&\"\"%F(!\"\"**F'F(F%F()F*\"\"$F(I\"zGF&F(F,*&)F *F'F()F0F'F(F,*(F%F(F2F(F0F(F(*&F*F(F3F(F(*&,&F%F,*$)F%F/F(F(F(F2F(F(* (,&*$F$F(F(F(F,F(F*F(F0F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"# \"\"\"I\"xG6\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*(\" \"#\"\"\"I\"xG6\"F%)I\"yGF'F$F%!\"\"*(F$F%F)F%I\"zGF'F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*$)I\"xG6\"\"\"#\"\"\"F(F(!\"\"I\"zGF&F(*(F'F(F %F()I\"yGF&F'F(F)*(F'F(F-F(F*F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "FF:=expand(subs(u=u[3],F)) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.5. P " }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "p:=5; m:=p; a:=x^3+10*x^2-43 2*x+5040; a mod p; u:=x; w:=x^2-2;\n" }{MPLTEXT 1 0 17 "e:=expand(a-u* w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\"xG6\"\"\"$ \"\"\"F(*&\"#5F()F%\"\"#F(F(*&\"$K%F(F%F(!\"\"\"%S]F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"$\"\"\"F(*&\"\"#F(F%F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ", (*&\"#5\"\"\")I\"xG6\"\"\"#F%F%*&\"$I%F%F'F%!\"\"\"%S]F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Gcdex(u,w,x,'s','t') mod p; s; t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\" #" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand( s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ", (*&\"\"#\"\"\")I\"xG6\"F$F%F%*&\"#')F%F'F%!\"\"\"%35F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"$\"\"\"F(*&\"\"#F()F%F*F(F(F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"!\"\"*&F'F(F% F(F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigm a,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,&I\"xG6\"\"\"\"\"\"#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expan d(w+sigma*m); e:=expand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"x G6\"\"\"#\"\"\"F(\"\"(!\"\"*&\"\"&F(F%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"$]%\"\"\"I\"xG6\"F%!\"\"\"%v]F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/ m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#=\"\"\"I\"xG6\"F%!\"\"\"$.#F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"!\"\"\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q *u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"!\"\"\"\"#\"\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+t au*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(\"#VF(*&\"#?F(F%F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"$D\"\"\"\"I\"xG6\"F%F%\"%]PF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$D\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c :=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 " " 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\")I\"xG6\"F$F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q *u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expand(w+sig ma*m); e:=expand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I \"xG6\"\"\"\"\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\" #\"\"\"F(\"$o\"F(*&\"#?F(F%F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$D'" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT 206 8 "E 6.6. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 " " {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "p:=5; m:= p; a:=x^4+1; a mod p; u:=x^2+2; w:=x^2-2; expand(u*w) mod p;\n" } {MPLTEXT 1 0 17 "e:=expand(a-u*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"%\"\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Gcdex (u,w,x,'s','t') mod p; s; t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=e xpand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=e xpand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\" \"#\"\"\"F(\"\"(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"# \"\"\"F(\"\"(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#]" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q *u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expand(w+sig ma*m); e:=expand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$ )I\"xG6\"\"\"#\"\"\"F(\"#dF(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\" xG6\"\"\"#\"\"\"F(\"#d!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%]K" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"$D\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mo d p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m ); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(\"$#=F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(\"$#=!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"&DJ$" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$D'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand(s*c) mod p; t au:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#`" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') \+ mod p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=exp and(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(\"%o5!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(\"%o5F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"(D19\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%DJ" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.7. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "p:=5; m:=p; a:=expand((2*x+5)*(6*x^2-10*x +7)); a mod p; u:=2*x; w:=x^2+2;\n" }{MPLTEXT 1 0 17 "e:=expand(a-u*w) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"#7\"\"\")I\"xG6\"\"\" $F%F%*&\"#5F%)F'\"\"#F%F%*&\"#OF%F'F%!\"\"\"#NF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"\"\"$F%F%F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"#5\"\"\")I\"xG6\"\"\"$F%F%*&F$F%)F'\"\"#F%F%*&\"#SF %F'F%!\"\"\"#NF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Gcdex(u ,w,x,'s','t') mod p; s; t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=ex pand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"#\"\"\")I\"xG6\"\"\"$F%F%*&F$F%)F'F$F%F%*&\"\")F%F'F%!\"\" \"\"(F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"#\"\"\")I\"xG6\"\"\"% F%F%*&F$F%)F'\"\"$F%F%*&F$F%)F'F$F%F%*&F$F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**$)I\"xG6\"\"\"$\"\"\"F(*$)F%\"\"#F(F(F%F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mo d p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ", &*&\"\"#\"\"\"I\"xG6\"F%!\"\"F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(* &\"\"#\"\"\")I\"xG6\"F$F%F%*&F$F%F'F%F%F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expa nd(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7\"\"\"I\"x G6\"F%F%\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\" \"\"F(\"\"$!\"\"*&\"#5F(F%F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\" $D\"\"\"\")I\"xG6\"\"\"#F%F%*&\"#]F%F'F%F%F+F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m ; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 " " {XPPMATH 20 ",(*&\"\"&\"\"\")I\"xG6\"\"\"#F%F%*&F)F%F'F%F%F)F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%F%*&F$F%F'F% F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\" #\"\"\"I\"xG6\"F%F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a -u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7\"\"\"I\"xG6\" F%F%\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F( \"#AF(*&\"#SF(F%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"$+&\"\"\") I\"xG6\"\"\"#F%!\"\"*&\"%+:F%F'F%F*\"$D'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$D\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e /m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"%\"\"\")I\"xG6\"\"\"#F%!\"\"*&\"#7F%F'F%F*\" \"&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"$\"\"\"F(*&\" \"#F()F%F*F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\" xG6\"F$F%!\"\"F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma :=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%!\"\"F%F(" }}{PARA 11 " " 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=ex pand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7\"\"\"I\"xG6\"F%F%\"#IF%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(\"$.\"!\"\"*& \"$5#F(F%F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"%+D\"\"\")I\"xG6 \"\"\"#F%F%*&\"%+vF%F'F%F%\"%DJF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$ D'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand (s*c) mod p; tau:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,(*&\"\"%\"\"\")I\"xG6\"\"\"#F%F%*&\"#7F%F'F%F%\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"$\"\"\"!\"\"*&\"\"#F()F%F+F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%F%F'F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mo d p; q; tau:=expand(tau+q*u) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ", &*&\"\"#\"\"\"I\"xG6\"F%F%F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"x G6\"!\"\"\"\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "u:=expand(u+tau*m); w:=expan d(w+sigma*m); e:=expand(a-u*w); m:=m*p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7\"\"\"I\"xG6\"F%F%\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,(*$)I\"xG6\"\"\"#\"\"\"F(\"$A&F(*&\"%S5F(F%F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"&+D\"\"\"\")I\"xG6\"\"\"#F%!\"\"*&\"&+v$F%F'F%F*\"& Dc\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"%DJ" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.8. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "p:=5; \+ m:=p; a:=expand((2*x+5)*(6*x^2-10*x+7));\n" }{MPLTEXT 1 0 28 "a mod p; u:=2*x; w:=x^2+2;\n" }{MPLTEXT 1 0 49 "alpha:=lcoeff(a); mu:=lcoeff(u ); nu:=lcoeff(w);\n" }{MPLTEXT 1 0 56 "aa:=alpha*a; u:=alpha*u/mu mod \+ m; w:=alpha*w/nu mod m;\n" }{MPLTEXT 1 0 18 "e:=expand(aa-u*w);" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"#7\"\"\")I\"xG6\"\"\"$F%F %*&\"#5F%)F'\"\"#F%F%*&\"#OF%F'F%!\"\"\"#NF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"\"\"$F%F%F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*$)I\"xG6\"\"\"#\"\"\"F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"$W\" \"\"\")I\"xG6\"\"\"$F%F%*&\"$?\"F%)F'\"\"#F%F%*&\"$K%F%F'F%!\"\"\"$?%F %" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%F%F%!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"$S\"\"\"\")I\"xG6\"\"\"$F%F%*&\"$ ?\"F%)F'\"\"#F%F%*&\"$I%F%F'F%!\"\"\"$?%F%" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "Gcdex(u,w,x,'s','t') mod p; s; t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"xG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mo d p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"#G\"\"\")I\"xG6\"\"\"$F%F% *&\"#CF%)F'\"\"#F%F%*&\"#')F%F'F%!\"\"\"#%)F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"#\"\"\")I\"xG6\"\"\"%F%!\"\"*$)F'\"\"$F%F**$)F'F$ F%F*F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"\"#\"\"\")I\"xG6\"\"\" $F%F%*$)F'F$F%F%F'F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"!\"\"*&F'F(F%F(F(F(F)" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"!\"\"\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "u:=expand(u+tau*m); w:=expand(w+sig ma*m); m:=m*p;\n" }{MPLTEXT 1 0 31 "mu:=lcoeff(u); nu:=lcoeff(w);\n" } {MPLTEXT 1 0 43 "u:=alpha*u/mu mod m; w:=alpha*w/nu mod m;\n" } {MPLTEXT 1 0 18 "e:=expand(aa-u*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 ", &*&\"\"$\"\"\"I\"xG6\"F%!\"\"\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,(*&\"\"#\"\"\")I\"xG6\"F$F%F%\"\"'!\"\"*&\"\"&F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ", &*&\"#7\"\"\"I\"xG6\"F%F%\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*& \"#7\"\"\")I\"xG6\"\"\"#F%F%\"#6!\"\"*&\"\"&F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"$D$\"\"\"I\"xG6\"F%!\"\"\"$v%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "c:=e/m; sigma:=expand(s*c) mod p; t au:=expand(t*c) mod p;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#8\"\"\" I\"xG6\"F%!\"\"\"#>F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\" )I\"xG6\"F$F%F%F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\" \"I\"xG6\"F%!\"\"F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "si gma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p;" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"!\"\"\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "u:=expand(u+tau*m); w:=expan d(w+sigma*m); m:=m*p;\n" }{MPLTEXT 1 0 31 "mu:=lcoeff(u); nu:=lcoeff(w );\n" }{MPLTEXT 1 0 43 "u:=alpha*u/mu mod m; w:=alpha*w/nu mod m;\n" } {MPLTEXT 1 0 18 "e:=expand(aa-u*w);" }}{PARA 11 "" 1 "" {XPPMATH 20 ", &*&\"#7\"\"\"I\"xG6\"F%F%\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*& \"#7\"\"\")I\"xG6\"\"\"#F%F%\"#9F%*&\"#?F%F'F%!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"$D\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7 \"\"\"I\"xG6\"F%F%\"#IF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"#7\"\" \")I\"xG6\"\"\"#F%F%\"#9F%*&\"#?F%F'F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "mu:=i gcd(coeffs(u)); u:=u/mu;\n" }{MPLTEXT 1 0 29 "nu:=igcd(coeffs(w)); w:= w/nu;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%F%\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"'\"\"\")I \"xG6\"\"\"#F%F%*&\"#5F%F'F%!\"\"\"\"(F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "A 6.1. Algoritmus. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "replace_lc:=proc(a,x,alpha) \+ local aa,aalpha,t;\n" }{MPLTEXT 1 0 18 " aa:=expand(a);\n" }{MPLTEXT 1 0 29 " aalpha:=lcoeff(aa,x,'t');\n" }{MPLTEXT 1 0 36 " aa:=expand( aa-aalpha*t+alpha*t);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6%I\"aG6\"I\"xGF%I&alphaGF%6%I#aaGF%I'aalphaGF%I\"tGF%F %F%C%>F)-I'expandG%*protectedG6#F$>F*-I'lcoeffGF06%F)F&.F+>F)-F/6#,(F) \"\"\"*&F*F;F+F;!\"\"*&F'F;F+F;F;F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "UnivariateHensel:=proc(a,u1,w1,x,p,B,gamma)\n" } {MPLTEXT 1 0 51 " local aa,alpha,e,u,uu,w,ww,m,s,t,q,c,sigma,tau;\n" }{MPLTEXT 1 0 51 " aa:=expand(a); alpha:=lcoeff(aa); aa:=gamma*aa;\n" }{MPLTEXT 1 0 50 " uu:=expand(u1); uu:=uu/lcoeff(uu)*gamma mod p;\n" }{MPLTEXT 1 0 50 " ww:=expand(w1); ww:=ww/lcoeff(ww)*alpha mod p;\n" }{MPLTEXT 1 0 33 " Gcdex(uu,ww,x,'s','t') mod p;\n" }{MPLTEXT 1 0 57 " u:=replace_lc(uu,x,gamma); w:=replace_lc(ww,x,alpha);\n" } {MPLTEXT 1 0 28 " e:=expand(aa-u*w); m:=p;\n" }{MPLTEXT 1 0 33 " whi le e<>0 and m<2*B*gamma do\n" }{MPLTEXT 1 0 63 " c:=e/m; sigma:=exp and(s*c) mod p; tau:=expand(t*c) mod p;\n" }{MPLTEXT 1 0 39 " sigma :=Rem(sigma,ww,x,'q') mod p;\n" }{MPLTEXT 1 0 34 " tau:=expand(tau+ q*uu) mod p;\n" }{MPLTEXT 1 0 47 " u:=expand(u+tau*m); w:=expand(w+ sigma*m);\n" }{MPLTEXT 1 0 32 " e:=expand(aa-u*w); m:=m*p;\n" } {MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 15 " if e=0 then\n" }{MPLTEXT 1 0 49 " u:=u/igcd(coeffs(u)); w:=w/igcd(coeffs(w));\n" }{MPLTEXT 1 0 12 " [u,w];\n" }{MPLTEXT 1 0 17 " else FAIL fi;\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6)I\"aG6\"I#u1GF%I#w1GF %I\"xGF%I\"pGF%I\"BGF%I&gammaG%*protectedG60I#aaGF%I&alphaGF%I\"eGF%I \"uGF%I#uuGF%I\"wGF%I#wwGF%I\"mGF%I\"sGF%I\"tGF%I\"qGF%I\"cGF%I&sigmaG F%I$tauGF%F%F%C0>F.-I'expandGF,6#F$>F/-I'lcoeffGF,6#F.>F.*&F+\"\"\"F.F G>F2-F?6#F&>F2-I$modGF%6$*(F2FG-FC6#F2!\"\"F+FGF)>F4-F?6#F'>F4-FM6$*(F 4FG-FC6#F4FRF/FGF)-FM6$-I&GcdexG6$F,I(_syslibGF%6'F2F4F(.F6.F7F)>F1-I+ replace_lcGF%6%F2F(F+>F3-Fao6%F4F(F/>F0-F?6#,&F.FG*&F1FGF3FGFR>F5F)?(F %FGFGF%30F0\"\"!2F5,$*(\"\"#FGF*FGF+FGFGC+>F9*&F0FGF5FR>F:-FM6$-F?6#*& F6FGF9FGF)>F;-FM6$-F?6#*&F7FGF9FGF)>F:-FM6$-I$RemGFjn6&F:F4F(.F8F)>F;- FM6$-F?6#,&F;FG*&F8FGF2FGFGF)>F1-F?6#,&F1FG*&F;FGF5FGFG>F3-F?6#,&F3FG* &F:FGF5FGFGFfo>F5*&F5FGF)FG@%/F0F_pC%>F1*&F1FG-I%igcdGF,6#-I'coeffsGF, 6#F1FR>F3*&F3FG-Fcs6#-Ffs6#F3FR7$F1F3I%FAILGF,F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 8 "E 6.9. P" }{TEXT 206 12 "\303\251lda." }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "debug(UnivariateH ensel); debug(replace_lc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I1Univaria teHenselG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I+replace_lcG6\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "UnivariateHensel(a,2*x,2*x^2 -1,x,5,10000,2);" }}{PARA 9 "" 1 "" {TEXT 207 88 "\{--> enter Univaria teHensel, args = 12*x^3+10*x^2-36*x+35, 2*x, 2*x^2-1, x, 5, 10000, 2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"#7\"\"\")I\"xG6\"\"\"$F%F%*&\"#5 F%)F'\"\"#F%F%*&\"#OF%F'F%!\"\"\"#NF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&\"#C\"\"\")I\"xG6\"\"\"$F%F %*&\"#?F%)F'\"\"#F%F%*&\"#sF%F'F%!\"\"\"#qF%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*& \"\"#\"\"\")I\"xG6\"F$F%F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&* &\"\"#\"\"\")I\"xG6\"F$F%F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"\"" }}{PARA 9 "" 1 "" {TEXT 207 40 "\{--> enter replace_lc, args = 2 *x, x, 2" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 9 "" 1 "" {TEXT 207 53 "<-- exi t replace_lc (now in UnivariateHensel) = 2*x\}" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I\"xG6\"F%F%" }}{PARA 9 "" 1 "" {TEXT 207 45 "\{--> enter replace_lc, args = 2*x^2-1, x, 12" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7\"\"\")I \"xG6\"\"\"#F%F%F%!\"\"" }}{PARA 9 "" 1 "" {TEXT 207 58 "<-- exit repl ace_lc (now in UnivariateHensel) = 12*x^2-1\}" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"#7\"\"\")I\"xG6\"\"\"#F%F%F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"#?\"\"\")I\"xG6\"\"\"#F%F%*&\"#qF%F'F%!\"\"F+F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"%\"\"\")I\"xG6\"\"\"#F%F%*&\"#9F%F'F%!\"\"F+F%" }}{PARA 11 " " 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"$\"\"\"!\"\"*$)F%\"\"#F(F(F%F)" } }{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(F%!\"\"F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\" #\"\"\"I\"xG6\"F%F%\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"#7\" \"\")I\"xG6\"\"\"#F%F%\"#6!\"\"*&\"\"&F%F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"#]\"\"\")I\"xG6\"\"\"#F%!\"\"*&\"#vF%F'F%F*\"$D\"F% " }}{PARA 11 "" 1 "" {XPPMATH 20 "\"#D" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"#\"\"\")I\"xG6\"F$F%!\"\"*&\"\"$F%F'F%F)\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"\"\"$F%!\"\"*&F$F%)F'F$F% F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\")I\"xG6\"F$F%F%*&F$ F%F'F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"!\"\"\"\"\"F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%F%\"\"&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ", (*&\"#7\"\"\")I\"xG6\"\"\"#F%F%\"#9F%*&\"#?F%F'F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"$D\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%F%\"\"&F%" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*&\"\"'\"\"\")I\"xG6\"\"\"#F%F%*&\"#5F %F'F%!\"\"\"\"(F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*&\"\"#\"\"\"I \"xG6\"F&F&\"\"&F&,(*&\"\"'F&)F'F%F&F&*&\"#5F&F'F&!\"\"\"\"(F&" }} {PARA 9 "" 1 "" {TEXT 207 70 "<-- exit UnivariateHensel (now at top le vel) = [2*x+5, 6*x^2-10*x+7]\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&*& \"\"#\"\"\"I\"xG6\"F&F&\"\"&F&,(*&\"\"'F&)F'F%F&F&*&\"#5F&F'F&!\"\"\" \"(F&" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 9 "E 6.10. P" }{TEXT 206 12 "\303\251lda." } }{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "p:=5; l:=1; a:=x^2*y^4*z-x*y^9*z^2+x*y*z^3+2*x-y^6*z^4-2*y^5*z; \n" }{MPLTEXT 1 0 24 "subs(y=1,z=1,a) mod p^l;" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*()I\"xG6\"\"\"#\"\"\")I\"yGF&\"\"%F(I\"zGF&F( F(*(F%F()F*\"\"*F()F,F'F(!\"\"*(F%F(F*F()F,\"\"$F(F(*&F'F(F%F(F(*&)F* \"\"'F()F,F+F(F1*(F'F()F*\"\"&F(F,F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(*&F'F(F%F(F(F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "u[1]:=x-2; w[1]:=x-1; expand(u[1]*w[1]) mod p^l;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(*&F'F(F%F(F(F'F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "aa:=expand(subs(y=Y+1,z=Z+1,a)) mod p^l;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",fr\"\"#\"\"\"*&F#F$I\"xG6\"F$F $*$)F&F#F$F$I\"YGF'!\"\"I\"ZGF'F+*(F)F$)F*\"\"%F$F,F$F$*(F)F$)F*\"\"$F $F,F$F+*(F)F$)F*F#F$F,F$F$*(F)F$F*F$F,F$F+*&F*F$F,F$F$*&F)F$F.F$F$*&F) F$F1F$F+*&F)F$F4F$F$*&F)F$F*F$F+*&F)F$F,F$F$*$)F,F#F$F+*&)F*\"\"'F$F=F $F+*&F?F$F,F$F$*&)F*\"\"&F$F=F$F+*&FCF$F,F$F+*&F*F$F=F$F+*(F#F$F&F$F*F $F$*&F&F$F,F$F$*&F&F$F1F$F$*&F&F$F4F$F+*(F#F$F&F$F=F$F$*&F&F$F.F$F+*&F &F$FCF$F+*&F&F$F?F$F$*&F&F$)F*\"\"(F$F+*&F&F$)F*\"\")F$F$*&F&F$)F*\"\" *F$F+*&F*F$)F,F2F$F$*&F&F$FYF$F$*&F#F$FCF$F$*$F?F$F+*$FYF$F$**F#F$F&F$ F1F$F,F$F$**F#F$F&F$F4F$F,F$F+*(F&F$FVF$F=F$F+**F#F$F&F$FVF$F,F$F+**F# F$F&F$FSF$F,F$F$*(F&F$FPF$F=F$F+**F#F$F&F$FPF$F,F$F+*(F&F$F?F$F=F$F$** F#F$F&F$F?F$F,F$F$*(F&F$FCF$F=F$F+**F#F$F&F$FCF$F,F$F+*(F&F$F.F$F=F$F+ *(F&F$F1F$F=F$F$*(F&F$F4F$F=F$F+*(F&F$F*F$F=F$F+**F#F$F&F$F.F$F,F$F+*( F&F$FSF$F=F$F$*(F&F$F*F$FYF$F$*&F?F$FYF$F$*&FCF$)F,F/F$F+*&FCF$FYF$F$* &F*F$F\\pF$F+*$F\\pF$F+*&F?F$F\\pF$F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "collect(aa,[Y,Z],`distributed`): aa:=sort(%,[Y,Z],tde g);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",\\p*(I\"xG6\"\"\"\")I\"YGF%\"\"* F&)I\"ZGF%\"\"#F&!\"\"**F,F&F$F&F'F&F+F&F-*(F$F&)F(\"\")F&F*F&F&*&)F( \"\"'F&)F+\"\"%F&F-*&F$F&F'F&F-**F,F&F$F&F0F&F+F&F&*(F$F&)F(\"\"(F&F*F &F-*&F3F&)F+\"\"$F&F&*&)F(\"\"&F&F5F&F-*&F$F&F0F&F&**F,F&F$F&F:F&F+F&F -*(,&F$F&F&F-F&F3F&F*F&F&*&F@F&F=F&F&*&F$F&F:F&F-*(,&*&F,F&F$F&F&F&F&F &F3F&F+F&F&*(,&F&F-F$F-F&F@F&F*F&F&*&FEF&F3F&F&*(,&FJF-F&F-F&F@F&F+F&F &*(F$F&)F(F6F&F*F&F-*&,&F$F-F,F&F&F@F&F&*(,&*$)F$F,F&F&FJF-F&FQF&F+F&F &*(F$F&)F(F>F&F*F&F&*&F(F&F5F&F-*&,&FVF&F$F-F&FQF&F&*(,&FJF&FVF-F&FYF& F+F&F&*(F$F&)F(F,F&F*F&F-*(,&F$F&F&F&F&F(F&F=F&F&*$F5F&F-*&,&FVF-F$F&F &FYF&F&*(FUF&FjnF&F+F&F&*(FLF&F(F&F*F&F&*&F\\oF&F=F&F&*&FfnF&FjnF&F&*( ,&F&F&FVF-F&F(F&F+F&F&*&,&F&F-FJF&F&F*F&F&*&,(FVF-FJF&F&F-F&F(F&F&*&,( FVF&F&F-F$F&F&F+F&F&FJF&FVF&F,F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "aa:=subs(Y=y-1,Z=z-1,aa) mod p^l;\n" }{MPLTEXT 1 0 204 "u[7]: =(x-2)+(-x+1)*(y-1)+(x-2)*(z-1)+x*(y-1)^2+(-x-2)*(y-1)*(z-1)+(-2)*(z-1 )^2+(-x)*(y-1)^3+x*(y-1)^2*(z-1)+(-2)*(y-1)*(z-1)^2+(z-1)^3+x*(y-1)^4+ (-x)*(y-1)^3*(z-1)+(y-1)*(z-1)^3+x*(y-1)^4*(z-1) mod p^l;\n" }{MPLTEXT 1 0 63 "w[7]:=(x-1)+(-1)*(z-1)+(-1)*(y-1)^5+(-1)*(y-1)^5*(z-1) mod p^ l;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",\\p\"\"#\"\"\"*&F#F$I\"xG6\"F$F$* $)F&F#F$F$**F#F$F&F$),&I\"yGF'F$F$!\"\"\"\")F$,&I\"zGF'F$F$F.F$F$*(F&F $F+F$)F0F#F$F$*(,&F&F$F$F$F$F,F$)F0\"\"$F$F$*&F&F$)F,\"\"*F$F.*&F&F$F+ F$F$*&F&F$)F,\"\"(F$F.*&,&F&F$F$F.F$)F,\"\"'F$F$*&,&F&F.F#F$F$)F,\"\"& F$F$*&,&F(F$F&F.F$)F,\"\"%F$F$*&,&F(F.F&F$F$)F,F7F$F$*&FHF$)F,F#F$F$*& ,(F(F.F%F$F$F.F$F,F$F$*&,(F(F$F$F.F&F$F$F0F$F$*&,&F$F.F%F$F$F3F$F$*&FA F$)F0FJF$F.*&FAF$F6F$F$*&FEF$FWF$F.*&FEF$F6F$F$*&F,F$FWF$F.*&F5F$F6F$F $*$FWF$F.*(,&F$F$F(F.F$F,F$F0F$F$*(F@F$FAF$F3F$F$*(,&F%F$F(F.F$FMF$F0F $F$*(F&F$F9F$F3F$F.**F#F$F&F$F9F$F0F$F.*(F&F$F=F$F3F$F.**F#F$F&F$F=F$F 0F$F.*(,&F%F$F$F$F$FAF$F0F$F$*(,&F$F.F&F.F$FEF$F3F$F$*(,&F%F.F$F.F$FEF $F0F$F$*(F&F$FIF$F3F$F.*(,&F(F$F%F.F$FIF$F0F$F$*(F&F$FMF$F3F$F$*(F&F$F OF$F3F$F.*(FioF$FOF$F0F$F$*(FdoF$F,F$F3F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",@I\"xG6\"\"\"\"\"\"#!\"\"*&,&F#F'F%F%F%,&I\"yGF$F%F%F'F% F%*&,&F#F%F&F'F%,&I\"zGF$F%F%F'F%F%*&F#F%)F*F&F%F%*(,&F#F'F&F'F%F*F%F. F%F%*&F&F%)F.F&F%F'*&F#F%)F*\"\"$F%F'*(F#F%F1F%F.F%F%*(F&F%F*F%F5F%F'* $)F.F8F%F%*&F#F%)F*\"\"%F%F%*(F#F%F7F%F.F%F'*&F*F%FF%F.F% F%" }}{PARA 11 "" 1 "" {XPPMATH 20 ",*I\"xG6\"\"\"\"I\"zGF$!\"\"*$),&I \"yGF$F%F%F'\"\"&F%F'*&F)F%,&F&F%F%F'F%F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "expand(u[7]) mod p^l; expand(w[7]) mod p^l; expand( aa-u[7]*w[7]) mod p^l;" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(\"\"#\"\"\"* (I\"xG6\"F$)I\"yGF'\"\"%F$I\"zGF'F$F$*&F)F$)F+\"\"$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"*&)I\"yGF$\"\"&F%I\"zGF$F%!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "A 6.2. Algoritmus. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "MultivariateDiophant:=proc(a ,c,E,d,p,k)\n" }{MPLTEXT 1 0 65 " local sigma,r,nu,i,A,aa,b,cc,EE,e,m onom,m,x,y,ee,cm,ds,alpha;\n" }{MPLTEXT 1 0 28 " r:=nops(a); nu:=nops (E);\n" }{MPLTEXT 1 0 16 " if nu>1 then\n" }{MPLTEXT 1 0 41 " x:=o p(1,E[nu]); alpha:=op(2,E[nu]);\n" }{MPLTEXT 1 0 26 " A:=mul(a[i],i =1..r);\n" }{MPLTEXT 1 0 36 " for i to r do b[i]:=A/a[i] od;\n" } {MPLTEXT 1 0 24 " aa:=subs(E[nu],a);\n" }{MPLTEXT 1 0 24 " cc:=s ubs(E[nu],c);\n" }{MPLTEXT 1 0 21 " EE:=E[1..nu-1];\n" }{MPLTEXT 1 0 50 " sigma:=MultivariateDiophant(aa,cc,EE,d,p,k);\n" }{MPLTEXT 1 0 55 " e:=mods(expand(c-add(sigma[i]*b[i],i=1..r)),p^k);\n" } {MPLTEXT 1 0 15 " monom:=1;\n" }{MPLTEXT 1 0 30 " for m to d whi le e<>0 do\n" }{MPLTEXT 1 0 31 " monom:=monom*(x-alpha);\n" } {MPLTEXT 1 0 2 " " }{MPLTEXT 1 0 22 " ee:=diff(e,[x$m]);" } {MPLTEXT 1 0 32 "\n cm:=subs(x=alpha,ee)/m!;" }{MPLTEXT 1 0 2 "\n " }{MPLTEXT 1 0 21 " if cm<>0 then\n" }{MPLTEXT 1 0 51 " d s:=MultivariateDiophant(aa,cm,EE,d,p,k);\n" }{MPLTEXT 1 0 66 " \+ for i to r do sigma[i]:=expand(sigma[i]+ds[i]*monom) od;\n" }{MPLTEXT 1 0 62 " e:=mods(expand(e-add(ds[i]*monom*b[i],i=1..r)),p^k);\n " }{MPLTEXT 1 0 11 " fi;\n" }{MPLTEXT 1 0 10 " od; \n" } {MPLTEXT 1 0 8 " else\n" }{MPLTEXT 1 0 14 " x:=E[1];\n" }{MPLTEXT 1 0 24 " sigma:=[0$i=1..r];\n" }{MPLTEXT 1 0 26 " for m from 0 t o d do\n" }{MPLTEXT 1 0 25 " cm:=coeff(c,x,m);\n" }{MPLTEXT 1 0 21 " if cm<>0 then\n" }{MPLTEXT 1 0 44 " ds:=UnivariateDio phant(a,x,m,p,k);\n" }{MPLTEXT 1 0 63 " for i to r do sigma[i]: =expand(sigma[i]+ds[i]*cm) od;\n" }{MPLTEXT 1 0 11 " fi;\n" } {MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 7 " fi;\n" }{MPLTEXT 1 0 36 " map((x,y)->mods(x,y),sigma,p^k);\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6(I\"aG6\"I\"cGF%I\"EGF%I\"dGF%I\"pGF%I\"kGF %64I&sigmaGF%I\"rGF%I#nuGF%I\"iGF%I\"AGF%I#aaGF%I\"bGF%I#ccGF%I#EEGF%I \"eGF%I&monomGF%I\"mGF%I\"xGF%I\"yGF%I#eeGF%I#cmGF%I#dsGF%I&alphaGF%F% F%C&>F--I%nopsG%*protectedG6#F$>F.-FA6#F'@%2\"\"\"F.C->F8-I#opGFB6$FI& F'6#F.>F=-FM6$\"\"#FO>F0-I$mulGFB6$&F$6#F//F/;FIF-?(F/FIFIF-I%trueGFB> &F2FZ*&F0FIFY!\"\">F1-I%subsGFB6$FOF$>F3-F_o6$FOF&>F4&F'6#;FI,&F.FIFIF \\o>F,-I5MultivariateDiophantGF%6(F1F3F4F(F)F*>F5-I%modsGFB6$-I'expand GFB6#,&F&FI-I$addGFB6$*&&F,FZFIFjnFIFenF\\o)F)F*>F6FI?(F7FIFIF(0F5\"\" !C&>F6*&F6FI,&F8FIF=F\\oFI>F:-I%diffGFB6$F57#-I\"$GFB6$F8F7>F;*&-F_o6$ /F8F=F:FI-I*factorialGFB6#F7F\\o@$0F;F^qC%>F<-F[p6(F1F;F4F(F)F*?(F/FIF IF-Fhn>Fip-Fbp6#,&FipFI*&&FF5-F_p6$-Fbp6#,&F5FI-Ffp6$*(F_s FIF6FIFjnFIFenF\\oFjpC%>F8&F'6#FI>F,7#-Fiq6$F^qFen?(F7F^qFIF(FhnC$>F;- I&coeffGFB6%F&F8F7@$FdrC$>F<-I3UnivariateDiophantGF%6'F$F8F7F)F*?(F/FI FIF-Fhn>Fip-Fbp6#,&FipFI*&F_sFIF;FIFI-I$mapGFB6%f*6$F8F9F%6$I)operator GF%I&arrowGF%F%-F_pFguF%F%F%F,FjpF%F%F%" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "A 6.3. Algoritmus. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "UnivariateDiophant:=proc(a,x ,m,p,k)\n" }{MPLTEXT 1 0 26 " local i,sigma,r,s,R,q;\n" }{MPLTEXT 1 0 15 " r:=nops(a);\n" }{MPLTEXT 1 0 15 " if r>2 then\n" }{MPLTEXT 1 0 42 " s:=MultiTermEEAlift(a,x,p,k); R:=[];\n" }{MPLTEXT 1 0 65 " \+ for i to r do R:=[op(R),mods(rem(x^m*s[i],a[i],x),p^k)] od;\n" } {MPLTEXT 1 0 8 " else\n" }{MPLTEXT 1 0 34 " s:=EEAlift(a[2],a[1],x ,p,k);\n" }{MPLTEXT 1 0 40 " q:=mods(quo(x^m*s[1],a[1],x),p^k);\n" }{MPLTEXT 1 0 8 " R:=[" }{MPLTEXT 1 0 42 "mods(expand(x^m*s[1]-q*a[ 1]),p^k),\n " }{MPLTEXT 1 0 33 "mods(expand(x^m*s[2]+q*a[2]),p^k) " }{MPLTEXT 1 0 4 "];\n" }{MPLTEXT 1 0 10 " fi; R;\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6'I\"aG6\"I\"xGF%I\"mGF%I\" pGF%I\"kGF%6(I\"iGF%I&sigmaGF%I\"rGF%I\"sGF%I\"RGF%I\"qGF%F%F%C%>F--I% nopsG%*protectedG6#F$@%2\"\"#F-C%>F.-I1MultiTermEEAliftGF%6&F$F&F(F)>F /7\"?(F+\"\"\"FBF-I%trueGF5>F/7$-I#opGF56#F/-I%modsGF56$-I$remGF%6%*&) F&F'FB&F.6#F+FB&F$FRF&)F(F)C%>F.-I(EEAliftGF%6'&F$6#F9&F$6#FBF&F(F)>F0 -FJ6$-I$quoG6$F5I(_syslibGF%6%*&FPFB&F.FgnFBFfnF&FT>F/7$-FJ6$-I'expand GF56#,&F`oFB*&F0FBFfnFB!\"\"FT-FJ6$-Fgo6#,&*&FPFB&F.FenFBFB*&F0FBFZFBF BFTF/F%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "MultiTermEEA lift:=proc(a,x,p,k) local i,r,s,beta,sigma;\n" }{MPLTEXT 1 0 30 " r:= nops(a); s:=[0$i=1..r];\n" }{MPLTEXT 1 0 17 " s[r-1]:=a[r];\n" } {MPLTEXT 1 0 64 " for i from r-2 by -1 to 1 do s[i]:=expand(a[i+1]*s[ i+1]) od;\n" }{MPLTEXT 1 0 12 " beta:=1;\n" }{MPLTEXT 1 0 19 " for i to r-1 do\n" }{MPLTEXT 1 0 62 " sigma:=MultivariateDiophant([s[i], a[i]],beta,[x],0,p,k);\n" }{MPLTEXT 1 0 37 " beta:=sigma[1]; s[i]:= sigma[2];\n" }{MPLTEXT 1 0 19 " od; s[r]:=beta;\n" }{MPLTEXT 1 0 6 " \+ s;\n" }{MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6&I\"a G6\"I\"xGF%I\"pGF%I\"kGF%6'I\"iGF%I\"rGF%I\"sGF%I%betaGF%I&sigmaGF%F%F %C*>F+-I%nopsG%*protectedG6#F$>F,7#-I\"$GF36$\"\"!/F*;\"\"\"F+>&F,6#,& F+F=F=!\"\"&F$6#F+?(F*,&F+F=\"\"#FBFBF=I%trueGF3>&F,6#F*-I'expandGF36# *&&F$6#,&F*F=F=F=F=&F,FQF=>F-F=?(F*F=F=FAFHC%>F.-I5MultivariateDiophan tGF%6(7$FJ&F$FKF-7#F&F:F'F(>F-&F.6#F=>FJ&F.6#FG>&F,FDF-F,F%F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "EEAlift:=proc(a,b,x,p,k) loc al ap,bp,s,t,sp,tp,i,m,e,c,q,sigma,tau;\n" }{MPLTEXT 1 0 33 " ap:=mod s(a,p); bp:=mods(b,p);\n" }{MPLTEXT 1 0 35 " mods(Gcdex(ap,bp,x,'s',' t'),p);\n" }{MPLTEXT 1 0 39 " sp:=mods(s,p); tp:=mods(t,p); m:=p;\n" }{MPLTEXT 1 0 19 " for i to k-1 do\n" }{MPLTEXT 1 0 43 " e:=expand (1-s*a-t*b); c:=mods(e/m,p);\n" }{MPLTEXT 1 0 61 " sigma:=mods(expa nd(sp*c),p); tau:=mods(expand(tp*c),p);\n" }{MPLTEXT 1 0 33 " q:=mo ds(Quo(sigma,bp,x),p);\n" }{MPLTEXT 1 0 40 " sigma:=mods(expand(sig ma-q*bp),p);\n" }{MPLTEXT 1 0 36 " tau:=mods(expand(tau+q*ap),p);\n " }{MPLTEXT 1 0 47 " s:=expand(s+sigma*m); t:=expand(t+tau*m);\n" } {MPLTEXT 1 0 13 " m:=m*p;\n" }{MPLTEXT 1 0 14 " od; [s,t];\n" } {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6'I\"aG6\"I\"b GF%I\"xGF%I\"pGF%I\"kGF%6/I#apGF%I#bpGF%I\"sGF%I\"tGF%I#spGF%I#tpGF%I \"iGF%I\"mGF%I\"eGF%I\"cGF%I\"qGF%I&sigmaGF%I$tauGF%F%F%C*>F+-I%modsG% *protectedG6$F$F(>F,-F;6$F&F(-F;6$-I&GcdexG6$FF/-F;6$F-F(>F0-F;6$F.F(>F2F(?(F1\"\"\"FR,&F)FRFR!\"\"I%trueGFF3-I'expandGF<6#,(FRFR*&F-FRF$FRFT*&F.FRF&FRFT>F4-F;6$*&F3FRF2FTF(>F6 -F;6$-FY6#*&F/FRF4FRF(>F7-F;6$-FY6#*&F0FRF4FRF(>F5-F;6$-I$QuoGFE6%F6F, F'F(>F6-F;6$-FY6#,&F6FR*&F5FRF,FRFTF(>F7-F;6$-FY6#,&F7FR*&F5FRF+FRFRF( >F--FY6#,&F-FR*&F6FRF2FRFR>F.-FY6#,&F.FR*&F7FRF2FRFR>F2*&F2FRF(FR7$F-F .F%F%F%" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 2 0 "" "%#%?G" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT 206 19 "A 6.4. Algoritmus. " }}{PARA 0 "" 0 "" {TEXT 201 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Multivariat eHensel:=proc(a,E,p,l,u,lcU)\n" }{MPLTEXT 1 0 74 " local nu,A,i,x,alp ha,U,UU,n,monom,maxdeg,aa,e,ee,co,oco,t,xx,m,j,c,dU;\n" }{MPLTEXT 1 0 18 " aa:=expand(a);\n" }{MPLTEXT 1 0 44 " nu:=nops(E); A:=[0$i=1..nu ]; n:=nops(u);\n" }{MPLTEXT 1 0 26 " A[nu]:=aa; maxdeg:=-1;\n" } {MPLTEXT 1 0 31 " for i from nu by -1 to 2 do\n" }{MPLTEXT 1 0 39 " \+ x:=op(1,E[i]); alpha:=op(2,E[i]);\n" }{MPLTEXT 1 0 30 " A[i-1]:=s ubs(E[i],A[i]);\n" }{MPLTEXT 1 0 56 " if degree(a,x)>maxdeg then ma xdeg:=degree(a,x) fi;\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 19 " \+ U:=u; xx:=E[1];\n" }{MPLTEXT 1 0 25 " for i from 2 to nu do\n" } {MPLTEXT 1 0 22 " UU:=U; monom:=1;\n" }{MPLTEXT 1 0 39 " x:=op(1 ,E[i]); alpha:=op(2,E[i]);\n" }{MPLTEXT 1 0 19 " for m to n do\n" } {MPLTEXT 1 0 25 " if lcU[m]<>1 then\n" }{MPLTEXT 1 0 48 " \+ co:=mods(subs(E[i+1..nu],lcU[m]),p^l);\n" }{MPLTEXT 1 0 47 " oc o:=lcoeff(collect(U[m],xx),xx,'t');\n" }{MPLTEXT 1 0 40 " U[m]: =expand(U[m]-oco*t+co*t);\n" }{MPLTEXT 1 0 11 " fi;\n" }{MPLTEXT 1 0 9 " od;\n" }{MPLTEXT 1 0 39 " e:=expand(A[i]-mul(U[j],j=1..n ));\n" }{MPLTEXT 1 0 43 " for j to degree(A[i],x) while e<>0 do\n" }{MPLTEXT 1 0 3 " " }{MPLTEXT 1 0 26 " monom:=monom*(x-alpha);" } {MPLTEXT 1 0 41 "\n c:=subs(E[i],diff(e,[x$j]))/j!;\n" }{MPLTEXT 1 0 20 " if c<>0 then\n" }{MPLTEXT 1 0 62 " dU:=Multivaria teDiophant(UU,c,E[1..i-1],maxdeg,p,l);\n" }{MPLTEXT 1 0 68 " fo r m to n do U[m]:=mods(expand(U[m]+dU[m]*monom),p^l) od;\n" }{MPLTEXT 1 0 53 " e:=mods(expand(A[i]-mul(U[m],m=1..n)),p^l);\n" } {MPLTEXT 1 0 9 " fi;" }{MPLTEXT 1 0 2 "\n" }{MPLTEXT 1 0 9 " o d;\n" }{MPLTEXT 1 0 7 " od;\n" }{MPLTEXT 1 0 54 " if a=expand(mul(U[ m],m=1..n)) then U else FAIL fi;\n" }{MPLTEXT 1 0 6 "end; " }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6(I\"aG6\"I\"EGF%I\"pGF%I\"lGF%I\"uGF%I$lcUG F%67I#nuGF%I\"AGF%I\"iGF%I\"xGF%I&alphaGF%I\"UGF%I#UUGF%I\"nGF%I&monom GF%I'maxdegGF%I#aaGF%I\"eGF%I#eeGF%I#coGF%I$ocoGF%I\"tGF%I#xxGF%I\"mGF %I\"jGF%I\"cGF%I#dUGF%F%F%C->F6-I'expandG%*protectedG6#F$>F,-I%nopsGFE 6#F&>F-7#-I\"$GFE6$\"\"!/F.;\"\"\"F,>F3-FI6#F)>&F-6#F,F6>F5!\"\"?(F.F, Fen\"\"#I%trueGFEC&>F/-I#opGFE6$FS&F&6#F.>F0-F\\o6$FgnF^o>&F-6#,&F.FSF SFen-I%subsGFE6$F^o&F-F_o@$2F5-I'degreeGFE6$F$F/>F5F]p>F1F)>F<&F&6#FS? (F.FgnFSF,FhnC)>F2F1>F4FSFjnF`o?(F=FSFSF3Fhn@$0&F*6#F=FSC%>F9-I%modsGF E6$-Fho6$&F&6#;,&F.FSFSFSF,F\\q)F'F(>F:-I'lcoeffGFE6%-I(collectG6$FEI( _syslibGF%6$&F1F]qFFcr-FD6#,(FcrFS*&F:FSF;FSFen*&F9FSF;FSFS>F7- FD6#,&FjoFS-I$mulGFE6$&F16#F>/F>;FSF3Fen?(F>FSFS-F^p6$FjoF/0F7FPC%>F4* &F4FS,&F/FSF0FenFS>F?*&-Fho6$F^o-I%diffGFE6$F77#-FN6$F/F>FS-I*factoria lGFEFcsFen@$0F?FPC%>F@-I5MultivariateDiophantGF%6(F2F?&F&6#;FSFfoF5F'F (?(F=FSFSF3Fhn>Fcr-Faq6$-FD6#,&FcrFS*&&F@F]qFSF4FSFSFiq>F7-Faq6$-FD6#, &FjoFS-F`s6$Fcr/F=FesFenFiq@%/F$-FD6#FcvF1I%FAILGFEF%F%F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "a; factor(a); collect(a,x); coeffs( %,x); gcd(%[1],%[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*()I\"xG6\"\" \"#\"\"\")I\"yGF&\"\"%F(I\"zGF&F(F(*(F%F()F*\"\"*F()F,F'F(!\"\"*(F%F(F *F()F,\"\"$F(F(*&F'F(F%F(F(*&)F*\"\"'F()F,F+F(F1*(F'F()F*\"\"&F(F,F(F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "*&,&I\"xG6\"\"\"\"*&)I\"yGF%\"\"&F&I \"zGF%F&!\"\"F&,(*()F)\"\"%F&F+F&F$F&F&*&F)F&)F+\"\"$F&F&\"\"#F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 ",**()I\"xG6\"\"\"#\"\"\")I\"yGF&\"\"%F(I \"zGF&F(F(*&,(*&F*F()F,\"\"$F(F(F'F(*&)F*\"\"*F()F,F'F(!\"\"F(F%F(F(*& )F*\"\"'F()F,F+F(F6*(F'F()F*\"\"&F(F,F(F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%,&*&)I\"yG6\"\"\"'\"\"\")I\"zGF'\"\"%F)!\"\"*(\"\"#F))F &\"\"&F)F+F)F-,(*&F&F))F+\"\"$F)F)F/F)*&)F&\"\"*F))F+F/F)F-*&)F&F,F)F+ F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "E:=[x,z=1,y=1]; subs(E[2..3],a);" }}{PARA 11 "" 1 " " {XPPMATH 20 "7%I\"xG6\"/I\"zGF$\"\"\"/I\"yGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*$)I\"xG6\"\"\"#\"\"\"F(*&F'F(F%F(F(\"\"$!\"\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "debug(MultivariateHensel);" }}{PARA 11 "" 1 "" {XPPMATH 20 "I3MultivariateHenselG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "MultivariateHensel(a,E,5,2,[x-1,x+3 ],[1,y^4*z]);" }}{PARA 9 "" 1 "" {TEXT 207 135 "\{--> enter Multivaria teHensel, args = x^2*y^4*z-x*y^9*z^2+x*y*z^3+2*x-y^6*z^4-2*y^5*z, [x, \+ z = 1, y = 1], 5, 2, [x-1, x+3], [1, y^4*z]" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*()I\"xG6\"\"\"#\"\"\")I\"yGF&\"\"%F(I\"zGF&F(F(*(F%F() F*\"\"*F()F,F'F(!\"\"*(F%F(F*F()F,\"\"$F(F(*&F'F(F%F(F(*&)F*\"\"'F()F, F+F(F1*(F'F()F*\"\"&F(F,F(F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"$" } }{PARA 11 "" 1 "" {XPPMATH 20 "7%\"\"!F#F#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*()I\"xG6\"\"\"# \"\"\")I\"yGF&\"\"%F(I\"zGF&F(F(*(F%F()F*\"\"*F()F,F'F(!\"\"*(F%F(F*F( )F,\"\"$F(F(*&F'F(F%F(F(*&)F*\"\"'F()F,F+F(F1*(F'F()F*\"\"&F(F,F(F1" } }{PARA 11 "" 1 "" {XPPMATH 20 "!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " I\"yG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&)I\"xG6\"\"\"#\"\"\"I\"zGF&F(F(*&F%F()F)F'F(!\"\"*&F% F()F)\"\"$F(F(*&F'F(F%F(F(*$)F)\"\"%F(F,*&F'F(F)F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " ,(*$)I\"xG6\"\"\"#\"\"\"F(*&F'F(F%F(F(\"\"$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"xG6\"\"\"\"F&!\"\",&F$F&\"\"$F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "I\"xG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"xG6\" \"\"\"F&!\"\",&F$F&\"\"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&\"\"$\"\"\"*&I \"zG6\"F$I\"xGF'F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",0*&I\"xG6\"\"\" \")I\"zGF%\"\"#F&!\"\"*&F$F&)F(\"\"$F&F&F$F**$)F(\"\"%F&F**&F)F&F(F&F* F-F&*&F(F&F$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"zG6\"\"\"\"F%! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"#\"\"\"I\"xG6\"F%F%\"\"' !\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$!\"\"\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 ",&I\"xG6\"\"\"\"I\"zGF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&*&I\"zG6\"\"\"\"I\"xGF%F&F&*&\"\"$F&F$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",.*&I\"xG6\"\"\"\")I\"zGF%\"\"$F&F&*&\"\"#F&F$F&F &*$)F(\"\"%F&!\"\"*&F+F&F(F&F/*(F)F&F(F&F$F&F/*&F)F&)F(F+F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "*$),&I\"zG6\"\"\"\"F'!\"\"\"\"#F'" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&*&\"\"$\"\"\"I\"xG6\"F%F%F$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"I\"zGF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**&I\"zG6\"\"\"\"I\"xGF%F&F&*&\"\"$F&F$F&!\"\"*&F)F&)F$ \"\"#F&F&F)F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",2*(\"\"$\"\"\"I\"xG6\" F%)I\"zGF'\"\"#F%!\"\"*&F&F%)F)F$F%F%F&F+*$)F)\"\"%F%F+F)F%*(F$F%F)F%F &F%F%*&F$F%F(F%F+*&F$F%F-F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&I \"zG6\"\"\"\"F'!\"\"\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6 \"\"\"\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$\"\"!\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"I\"zGF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I\"zG6\"\"\"\"I\"xGF%F&F&\"\"#F&*$)F$\"\"$F& F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"xG6\"\"\"\"I\"zGF%!\"\",(*&F'F&F$F&F&\"\"#F&*$)F' \"\"$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"yG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "*&)I\"yG6\"\"\"%\"\"\"I\"zGF%F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "I\"zG6\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(\"\"#\"\"\"*$)I\"zG6\"\"\"$F$F$*()I\"yGF(\"\"%F$F'F$I\"xGF(F$F$" } }{PARA 11 "" 1 "" {XPPMATH 20 ",2*(I\"xG6\"\"\"\")I\"yGF%\"\"*F&)I\"zG F%\"\"#F&!\"\"*(F$F&F(F&)F+\"\"$F&F&*&)F(\"\"'F&)F+\"\"%F&F-*(F,F&)F( \"\"&F&F+F&F-*&F$F&F/F&F-*&F,F&F+F&F&*$F4F&F&*()F(F5F&F*F&F$F&F&" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&I\"yG6\"\"\"\"F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",**(\"\"&\"\"\"I\"xG6\"F%)I\"zGF'\"\"#F%!\"\"*&F&F%)F )\"\"$F%F%*&\"\"'F%)F)\"\"%F%F+*&\"#5F%F)F%F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,$*&\"\"&\"\"\"I\"zG6\"F&!\"\"*$)F'\"\"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(I\"xG6\"\"\"\"*&\"\"%F%I\"zGF$F%F%*(\"\"&F%I\"y GF$F%F(F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*()I\"yG6\"\"\"%\"\" \"I\"zGF&F(I\"xGF&F(F(*&F%F()F)\"\"$F(F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",4*(I\"xG6\"\"\"\")I\"yGF%\"\"*F&)I\"zGF%\"\"#F&!\"\"*&)F (\"\"'F&)F+\"\"%F&F-*(F,F&)F(\"\"&F&F+F&F-**F2F&)F(F2F&F*F&F$F&F-*(F2F &F(F&F1F&F-*&\"\")F&F+F&F-**F5F&F4F&F*F&F$F&F&*(F5F&)F(F,F&F1F&F&*(\"# 5F&F(F&F+F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&I\"yG6\"\"\"\"F'! \"\"\"\"#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*(\"#5\"\"\"I\"xG6\"F%) I\"zGF'\"\"#F%!\"\"*&F$F%)F)\"\"%F%F+*&\"#?F%F)F%F+" }}{PARA 11 "" 1 " " {XPPMATH 20 "7$,$*&\"#5\"\"\"I\"zG6\"F&!\"\"\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 ",*I\"xG6\"\"\"\"*&\"\"'F%I\"zGF$F%!\"\"*(\"#5F%I\"yGF$F %F(F%F)*(F+F%F(F%)F,\"\"#F%F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*()I \"yG6\"\"\"%\"\"\"I\"zGF&F(I\"xGF&F(F(*&F%F()F)\"\"$F(F(\"\"#F(" }} {PARA 11 "" 1 "" {XPPMATH 20 ",:*(I\"xG6\"\"\"\")I\"yGF%\"\"*F&)I\"zGF %\"\"#F&!\"\"*&)F(\"\"'F&)F+\"\"%F&F-*(F,F&)F(\"\"&F&F+F&F-**F0F&)F(F2 F&F*F&F$F&F&*(F0F&F(F&F1F&F&*&\"#7F&F+F&F&**\"#5F&F4F&F*F&F$F&F&*(FF&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&I\"yG6\"\"\"\" F'!\"\"\"\"$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*(\"$S#\"\"\"I\"xG6 \"F%)I\"zGF'\"\"#F%F%*&\"#5F%)F)\"\"%F%!\"\"*&\"#?F%F)F%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,$*&\"#5\"\"\"I\"zG6\"F&!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,I\"xG6\"\"\"\"*&\"\"%F%I\"zGF$F%F%*(\"#5F%I\"y GF$F%F(F%F%*(\"\"&F%F(F%)F+\"\"#F%!\"\"*(F*F%)F+\"\"$F%F(F%F0" }} {PARA 11 "" 1 "" {XPPMATH 20 ",(*()I\"yG6\"\"\"%\"\"\"I\"zGF&F(I\"xGF& F(F(*&F%F()F)\"\"$F(F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 ",@*(I\" xG6\"\"\"\")I\"yGF%\"\"*F&)I\"zGF%\"\"#F&!\"\"*&)F(\"\"'F&)F+\"\"%F&F- *(F,F&)F(\"\"&F&F+F&F-**F2F&)F(F2F&F*F&F$F&F-*(F2F&F(F&F1F&F-*&\"\")F& F+F&F-**\"#5F&F4F&F*F&F$F&F-*(FF&F&**FF&F*F&F/F&F$F&F&*(F>F&F1F&)F(\"\"$F&F&*(F5F&F+F &FF&F$F&)F(\"\"(F&F*F&F-*(F>F&F7F&F1F&F-*(F5F&FAF&F+F&F&**F5F &F$F&)F(\"\")F&F*F&F&*(F5F&F4F&F1F&F&*(F>F&F7F&F+F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "*$),&I\"yG6\"\"\"\"F'!\"\"\"\"&F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*&I\"xG6\"\"\"\")I\"zGF%\"\"#F&!\"\"*$)F(\"\"%F&F**&F) F&F(F&F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,$I\"zG6\"!\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ",&I\"xG6\"\"\"\"*&)I\"yGF$\"\"&F%I\"zGF$ F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",(*()I\"yG6\"\"\"%\"\"\"I\"zG F&F(I\"xGF&F(F(*&F%F()F)\"\"$F(F(\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"xG6\"\"\"\"*&)I\"yGF %\"\"&F&I\"zGF%F&!\"\",(*()F)\"\"%F&F+F&F$F&F&*&F)F&)F+\"\"$F&F&\"\"#F &" }}{PARA 9 "" 1 "" {TEXT 207 77 "<-- exit MultivariateHensel (now at top level) = [x-y^5*z, y^4*z*x+y*z^3+2]\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$,&I\"xG6\"\"\"\"*&)I\"yGF%\"\"&F&I\"zGF%F&!\"\",(*()F) \"\"%F&F+F&F$F&F&*&F)F&)F+\"\"$F&F&\"\"#F&" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 205 15 "7. 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