{VERSION 6 0 "SGI MIPS UNIX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Head ing 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 4 "" 0 "" {TEXT -1 18 "Calculations in k[" } {XPPEDIT 18 0 "x[1],x[2];" "6$&%\"xG6#\"\"\"&F$6#\"\"#" }{TEXT -1 5 ", ...," }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 49 "]/I where I is an ideal: finite dimensional case." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "interface(imaginaryunit=_i):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groebner):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "I := [x*y^3-x^2,x^3*y^2-y];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG7$,&*&%\"xG\"\"\")%\"yG\"\"$F)F)*$)F(\"\"#F)!\"\" ,&*&)F(F,F))F+F/F)F)F+F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "G := Basis(I,tdeg(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7&, &*&%\"xG\"\"\"%\"yGF)!\"\"*$)F*\"\"%F)F),&*&F(F))F*\"\"$F)F)*$)F(\"\"# F)F+,&*$)F(F.F)F)*$)F*F5F)F+,&*&)F(F2F)F:F)F)F*F+" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 41 "LTI := map(LeadingMonomial,G,tdeg(x,y));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LTIG7&*$)%\"yG\"\"%\"\"\"*&%\"xGF *)F(\"\"$F**$)F,F)F**&)F,F.F*)F(\"\"#F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "LTC := [1,x,x^2,x^3,y,x*y,x^2*y,x^3*y,y^2,y^2*x,y^2*x ^2,y^3];\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LTCG7.\"\"\"%\"xG*$)F '\"\"#F&*$)F'\"\"$F&%\"yG*&F'F&F.F&*&F.F&F)F&*&F,F&F.F&*$)F.F*F&*&F'F& F3F&*&F3F&F)F&*$)F.F-F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 " nops(\{op(LTC)\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := 2*x+3*y^2-2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*&\"\"#\"\"\"%\"xGF(F(*&\"\"$F()%\"yGF'F(F(F '!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "finv := add( c[i] *LTC[i], i=1..nops(LTC) );\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%fin vG,:&%\"cG6#\"\"\"F)*&&F'6#\"\"#F)%\"xGF)F)*&&F'6#\"\"$F))F.F-F)F)*&&F '6#\"\"%F))F.F2F)F)*&&F'6#\"\"&F)%\"yGF)F)*(&F'6#\"\"'F)F.F)F=F)F)*(&F '6#\"\"(F)F=F)F3F)F)*(&F'6#\"\")F)F8F)F=F)F)*&&F'6#\"\"*F))F=F-F)F)*(& F'6#\"#5F)F.F)FNF)F)*(&F'6#\"#6F)FNF)F3F)F)*&&F'6#\"#7F))F=F2F)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "zero := NormalForm( f*finv, \+ G, tdeg(x,y) ) - 1;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%zeroG,<*&\" \"#\"\"\"&%\"cG6#F(F(!\"\"*&,&*&F'F(&F*6#F'F(F,*&F'F(F)F(F(F(%\"xGF(F( *&,(*&F'F(&F*6#\"\"&F(F,*&F'F(&F*6#\"#6F(F(*&\"\"$F(&F*6#\"\"%F(F(F(% \"yGF(F(*&,(*&F'F(&F*6#\"#7F(F,*&F?F(F7F(F(*&F'F(&F*6#\"\")F(F(F()FCF? F(F(*&,**&F'F(&F*6#F?F(F,*&F'F(F0F(F(*&F'F(FGF(F(*&F?F(&F*6#\"\"'F(F(F ()F3F'F(F(*&,(*&F'F(F@F(F,*&F'F(FSF(F(*&F?F(&F*6#\"\"(F(F(F()F3F?F(F(* &,**&F'F(&F*6#\"\"*F(F,*&F?F(F)F(F(*&F'F(F@F(F(*&F?F(FLF(F(F()FCF'F(F( *(,**&F?F(FGF(F(*&F'F(FboF(F(*&F'F(&F*6#\"#5F(F,*&F?F(F0F(F(F(F3F(FhoF (F(*(,(*&F'F(FXF(F,*&F'F(F7F(F(*&F?F(FboF(F(F(F3F(FCF(F(*(,(*&F?F(F^pF (F(*&F'F(F[oF(F,*&F'F(FXF(F(F(FCF(FenF(F(*(,(*&F?F(F;F(F(*&F'F(FLF(F,* &F'F(F[oF(F(F(F^oF(FCF(F(*(,(*&F'F(F;F(F,*&F'F(F^pF(F(*&F?F(FSF(F(F(Fh oF(FenF(F(F(F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The main point \+ here is that if " }{XPPEDIT 18 0 "f;" "6#%\"fG" }{TEXT -1 17 " has an inverse " }{XPPEDIT 18 0 "`@@`(f,-1);" "6#-%#@@G6$%\"fG,$\"\"\"!\"\" " }{TEXT -1 7 " then " }{XPPEDIT 18 0 "f*`@@`(f,-1);" "6#*&%\"fG\"\" \"-%#@@G6$F$,$F%!\"\"F%" }{TEXT -1 9 " = 1 mod " }{XPPEDIT 18 0 "I;" " 6#%\"IG" }{TEXT -1 10 " . Since " }{XPPEDIT 18 0 "I;" "6#%\"IG" } {TEXT -1 100 " is finite dimensional and we know the complement of then we know the form of elements in " }{XPPEDIT 18 0 "I;" "6#% \"IG" }{TEXT -1 39 " so we can set up a linear system over " } {XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 27 ", which in this example is \+ " }{TEXT 256 1 "Q" }{TEXT -1 35 ", to solve for the coefficients of " }{XPPEDIT 18 0 "`@@`(f,-1);" "6#-%#@@G6$%\"fG,$\"\"\"!\"\"" }{TEXT -1 2 " ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqns := \{coeffs(z ero,[x,y])\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<.,(*&\"\"$\" \"\"&%\"cG6#\"#5F)F)*&\"\"#F)&F+6#\"\"(F)!\"\"*&F/F)&F+6#\"\"'F)F),(*& F/F)F5F)F3*&F/F)&F+6#\"\"&F)F)*&F(F)&F+6#\"\"*F)F),**&F(F)&F+6#\"#7F)F )*&F/F)F?F)F)*&F/F)F*F)F3*&F(F)&F+6#F/F)F),**&F/F)&F+6#F(F)F3*&F/F)FJF )F)*&F/F)FDF)F)*&F(F)F5F)F),**&F/F)F?F)F3*&F(F)&F+6#F)F)F)*&F/F)&F+6# \"\"%F)F)*&F(F)&F+6#\"\")F)F),(*&F(F)&F+6#\"#6F)F)*&F/F)FgnF)F3*&F/F)F 0F)F),(*&F/F)F\\oF)F3*&F/F)F*F)F)*&F(F)FNF)F),(*&F/F)FYF)F3*&F/F)FNF)F )*&F(F)F0F)F),(*&F/F)FDF)F3*&F(F)F;F)F)*&F/F)FgnF)F),(*&F/F)F;F)F3*&F/ F)F\\oF)F)*&F(F)FYF)F),&*&F/F)FJF)F3*&F/F)FVF)F),&*&F/F)FVF)F3F)F3" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "sols := solve(eqns);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%%solsG<./&%\"cG6#\"\"'#!%s\\\"&$y^/& F(6#\"#7#\"&.X(\"'mN5/&F(6#\"#6#\"%p#*F-/&F(6#\"#5#\"%;MF-/&F(6#\"\"*# !&CO\"F-/&F(6#\"\")#\"&6\"GF4/&F(6#\"\"(#\"$_\"F-/&F(6#\"\"&#\"&ka\"F- /&F(6#\"\"%#\"%ITF-/&F(6#\"\"$#\"%-RF-/&F(6#\"\"##!\"\"Fbo/&F(6#\"\"\" Fco" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Now check it by multiplica tion in the quotient ring." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "NormalForm( f*subs(sols,finv), G, tdeg(x,y) );\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "G2 := factor( Basis(I,plex(x,y)) );" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#>%#G2G7%*,%\"yG\"\"\",&F'F(F(!\"\"F(,&F'F(F(F(F(,,*$)F'\"\"%F(F(*$)F '\"\"$F(F(*$)F'\"\"#F(F(F'F(F(F(F(,,F-F(F0F*F3F(F'F*F(F(F(*&F'F(,&F0F* %\"xGF(F(*&F8F(,&F9F(F0F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Cl early y is a zero divisor in the quotient ring so it will not have an \+ inverse." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f := y;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG%\"yG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "zero := NormalForm( f*finv, G, tdeg(x,y) ) - 1:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqns := \{coeffs(zero,[x,y ])\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<-!\"\"&%\"cG6#\"\"$& F(6#\"\"%&F(6#\"\"&&F(6#\"\"'&F(6#\"\"(&F(6#\"\"*&F(6#\"#5&F(6#\"#6,&& F(6#\"\"#\"\"\"&F(6#\"#7FD,&&F(6#FDFD&F(6#\"\")FD" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "sols := solve(eqns);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG6\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "How many solutions are in V where V = " }{TEXT 257 1 "V" }{TEXT -1 5 "(I) ?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Gy := Groebner[Basis] (I,plex(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GyG7%,&*$)%\"yG\" #6\"\"\"F+F)!\"\",&*&%\"xGF+F)F+F+*$)F)\"\"%F+F,,&*$)F/\"\"#F+F+*$)F) \"\"'F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fy := Gy[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fyG,&*$)%\"yG\"#6\"\"\"F*F(!\"\" " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Gx := Groebner[Basis](I ,plex(y,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GxG7$,&*$)%\"xG\"#7 \"\"\"F+*$)F)\"\"#F+!\"\",&%\"yGF+*$)F)\"\"(F+F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "fx := Gx[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fxG,&*$)%\"xG\"#7\"\"\"F**$)F(\"\"#F*!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 115 "It follows that the variety of I is finite, and t here are at most 12 solutions for x and 11 for y hence |V| <= 132." }} {PARA 0 "" 0 "" {TEXT -1 241 "These univariate polynomials fx and fy w hich are in the ideal I can be found directly using the following comm and. One can compute them using the FGLM algorithm instead of using t he lex monomial ordering to eliminate all variables except 1." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Groebner[UnivariatePolynomia l](x,I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"#7\"\"\"F(*$)F &\"\"#F(!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Groebner[U nivariatePolynomial](y,I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\" yG\"#6\"\"\"F(F&!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "I'll cou nt the number of solutions by computing a prime decomposition of the r adical of I." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "with(Polyno mialIdeals):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "I := ;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"IG-%$<,>G6$,&*&%\"xG\"\"\")%\"yG \"\"$F+F+*$)F*\"\"#F+!\"\",&*&)F*F.F+)F-F1F+F+F-F2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "IsRadical(I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "The use of simp lify below is to compute Groebner bases for each prime component of J. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "map(Simplify,[PrimeDeco mposition(Radical(I))]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7'-%$<,>G6 $,&%\"yG\"\"\"F)F),&%\"xGF)F)F)-F%6$,,*$)F+\"\"%F)F)*$)F+\"\"$F)F)*$)F +\"\"#F)F)F+F)F)F),&F(F)F5!\"\"-F%6$,,F/F)F2F9F5F)F+F9F)F),&F(F)F5F)-F %6$F+F(-F%6$,&F(F)F)F9,&F+F)F)F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "From \+ here we see that there are 1 + 4 + 4 + 1 + 1 = 11 solutions. " }}}} {MARK "38 0 0" 66 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }