{VERSION 2 3 "DEC ALPHA UNIX" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 0 1 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 }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 0 0 0 6 6 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 }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 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item " 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "C ourier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 20 "The Groebner Package" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT 256 53 "by Frederic Chyzak, Algorithms Project, INRIA, France" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 17 "" 0 "" {TEXT -1 24 "Frederic.Chyzak@inria.fr" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{HYPERLNK 17 "Groebner" 2 "Groebner" "" }{TEXT -1 90 " package extends the funct ionality of the grobner package into several directions, namely:" }}} {EXCHG {PARA 15 "" 0 "" {TEXT -1 88 "calculations over more complicate d ground fields are now possible with a single package;" }}{PARA 15 " " 0 "" {TEXT -1 83 "new term orders are available, in view of new appl ications and improved efficiency;" }}{PARA 15 "" 0 "" {TEXT -1 72 "cal culations of common invariants of ideals and varieties are available; " }}{PARA 15 "" 0 "" {TEXT -1 49 "facilities for change of orderings a re available." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(Groeb ner);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75%%fglmG%'gbasisG%'gsolveG%+ hilbertdimG%,hilbertpolyG%.hilbertseriesG%-inter_reduceG%*is_finiteG%, is_solvableG%*leadcoeffG%(leadmonG%)leadtermG%(normalfG%/pretend_gbasi sG%'reduceG%&spolyG%*termorderG%*testorderG%)univpolyG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "Some syntactical differences exist betwe en the grobner and the Groebner packages. These differences are motiv ated by the extension of functionality, particularly in the number of \+ term orderings. Specifically, a call" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 30 "grobner[gbasis](G,[x,y],plex);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "becomes" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 30 "Groebner[gb asis](G,plex(x,y));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "while the \+ call" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 0 25 "grobner[gbasis](G,\{x,y\}) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "with " }{XPPEDIT 18 0 "tdeg" "I%tdegG6\"" }{TEXT -1 66 " being the standard has no counterpart in G roebner. For instance," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 " grobner[gbasis]([x^2-2*x*z+5,x*y^2+y*z^3,3*y^2-8*z^3],[y,x,z],plex);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7',&*$%\"yG\"\"#\"\"$*$%\"zGF(!\"),* *&F&\"\"\"F*F(\"#!)*$F*\"\")!\"$*$F*\"\"(\"#K*$F*\"\"&!#S,(*$%\"xGF'F. *&F;F.F*F.!\"#F7F.,*F3!#'*F0\"\"*F6\"$?\"*&F*F(F;F.\"$S',**$F*\"\"'\"$ S#F)\"%+;F0F?*$F*F@F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "becomes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Groebner[gbasis]([x^2-2*x *z+5,x*y^2+y*z^3,3*y^2-8*z^3],plex(y,x,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7',**$%\"zG\"\"'\"$S#*$F&\"\"$\"%+;*$F&\"\")!#'**$F&\" \"*F0,**$F&\"\"(F.F,F0*$F&\"\"&\"$?\"*&F&F*%\"xG\"\"\"\"$S',(*$F8\"\"# F9*&F8F9F&F9!\"#F5F9,**&%\"yGF9F&F*\"#!)F,!\"$F2\"#KF4!#S,&*$FBF=F*F)! \")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "Moreover, for complicated calculations, the Groebner package requires data structures that are \+ created by the " }{HYPERLNK 17 "Ore_algebra" 2 "Ore_algebra" "" } {TEXT -1 9 " package." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "wi th(Ore_algebra);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74%-Ore_to_DESolG% -Ore_to_RESolG%,Ore_to_diffG%-Ore_to_shiftG%-annihilatorsG%)applyoprG% -diff_algebraG%-poly_algebraG%/qshift_algebraG%/rand_skew_polyG%.shift _algebraG%-skew_algebraG%*skew_elimG%+skew_gcdexG%*skew_pdivG%+skew_po werG%*skew_premG%-skew_productG" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "Ground fields" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "The Groebner package supports transcendental and algebraic extensions of the field of rational numbers and finite fields. This includes algebraic numbe r fields. Here is a sample computation with complex numbers in " } {XPPEDIT 18 0 "Q*[I,sqrt(2)]*[x,y]" "*(%\"QG\"\"\"7$%\"IG-%%sqrtG6#\" \"#F$7$%\"xG%\"yGF$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "G:=[x^2+y^2-I,sqrt(2)*y+sqrt(2)*(x-sqrt(2))-1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG7$,(*$%\"xG\"\"#\"\"\"*$%\"yGF)F *%\"IG!\"\",(*&F)#F*F)F,F*F**&F)F1,&F(F**$F)F1F.F*F*F.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "GB:=gbasis(G,tdeg(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7$,(%\"yG\"\"#%\"xGF(*$F(#\"\"\"F(!\"$ ,**$F)F(\"\"%*&F(F+F)F,!\"'%\"IG!\"#\"\"*F," }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 128 "For computations with modular integers, one has to dec lare the corresponding algebra of polynomials. This is dealt with by \+ the " }{HYPERLNK 17 "Ore_algebra" 2 "Ore_algebra" "" }{TEXT -1 136 " p ackage. (Ore polynomials extend commutative polynomials, and the gene ral mechanism to deal with general algebras is in this package.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=poly_algebra(x,y,z,charac teristic=5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%,Ore_algebraG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "G:=[x^2-2*x*z+5,x*y^2+y*z ^3,3*y^2-8*z^3]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A[norma lizer](G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,&*$%\"xG\"\"#\"\"\"*& F&F(%\"zGF(\"\"$,&*&F&F(%\"yGF'F(*&F.F(F*F+F(,&*$F.F'F+*$F*F+F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "T:=termorder(A,plex(x,y,z)): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "gbasis(G,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7',&*$%\"zG\"\")\"\"\"*$F&\"\"(F(,&*$F&\"\"' \"\"$*&%\"yGF(F&\"\"%F(,&*$F0\"\"#F(*$F&F.F1,&*&F0F(F&F.F(*&F&F.%\"xGF (F(,&*$F9F4F(*&F9F(F&F(F." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Te rm orderings" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "The grobner packag e allows two types of term orderings:" }}}{EXCHG {PARA 15 "" 0 "" {XPPEDIT 18 0 "tdeg" "I%tdegG6\"" }{TEXT -1 31 ", for \"efficient\" no rmal forms," }}{PARA 15 "" 0 "" {XPPEDIT 18 0 "plex" "I%plexG6\"" } {TEXT -1 49 ", for \"slow\" triangularizations (system solving)." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 211 "There was a need for specialized \+ orderings for the purpose of efficient elimination and for term orderi ngs defined by a matrix for the purpose of generality. The Groebner p ackage implements both orderings, the " }{XPPEDIT 18 0 "lexdeg" "I'lex degG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "matrix" "I'matrixG6\"" } {TEXT -1 52 " types respectively, together with a more practical " } {XPPEDIT 18 0 "mdeg" "I%mdegG6\"" }{TEXT -1 68 " for weighted ordering s. User-defined orderings are also available." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "As an example, here is a calculation that is perform ed far faster using " }{XPPEDIT 18 0 "lexdeg" "I'lexdegG6\"" }{TEXT -1 12 " than using " }{XPPEDIT 18 0 "plex" "I%plexG6\"" }{TEXT -1 39 " . Assume we want to find the optima of" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "phi:=x^3+2*x*y*z-z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,(*$%\"xG\"\"$\"\"\"*(F'F)%\"yGF)%\"zGF)\"\"#*$F,F-!\"\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "on the sphere" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "const:=x^2+y^2+z^2-1;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%&constG,**$%\"xG\"\"#\"\"\"*$%\"yGF(F)*$%\"zGF (F)!\"\"F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "By Lagrange method, we compute" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "G:=\{seq(dif f(phi,v)-diff(const,v)*t,v=[x,y,z]),const\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG<&,(*$%\"xG\"\"#\"\"$*&%\"zG\"\"\"%\"yGF-F)*&F(F- %\"tGF-!\"#,&*&F(F-F,F-F)*&F.F-F0F-F1,*F'F-*$F.F)F-*$F,F)F-!\"\"F-,(*& F(F-F.F-F)F,F1*&F,F-F0F-F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "and we eliminate " }{XPPEDIT 18 0 "t" "I\"tG6\"" }{TEXT -1 40 " by the us e of an appropriate term order" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "GB:=gbasis(G,lexdeg([t],[x,y,z]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#GBG7*,**$%\"xG\"\"#\"\"\"*$%\"yGF)F**$%\"zGF)F*!\"\" F*,,*&F(F*F.F*\"#<*&F.F)F,F*F2*&F(F*F,F*F/F.\"#8*$F.\"\"$!#8,**&F,F)F. F*F2F6\"\"(F4!\"'F.!\"(,.*$F,F7F2F6\"#6F4F=F1!# " 0 "" {MPLTEXT 1 0 17 "remove(has,GB,t) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7),**$%\"xG\"\"#\"\"\"*$%\"yGF'F( *$%\"zGF'F(!\"\"F(,,*&F&F(F,F(\"#<*&F,F'F*F(F0*&F&F(F*F(F-F,\"#8*$F,\" \"$!#8,**&F*F'F,F(F0F4\"\"(F2!\"'F,!\"(,.*$F*F5F0F4\"#6F2F;F/!# " 0 "" {MPLTEXT 1 0 33 "GB2:=gbasis(%,lexdeg([x,y],[z]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$GB2G7),**$%\"zG\"\"(\"%_6*$F(\"\"&!%j<*$F (\"\"$\"$b'F(!#W,,*&F(\"\"\"%\"yGF4\"%qw*&%\"xGF4F(\"\"#F6*$F(\"\"'\"& wD(*$F(\"\"%!'Lr5*$F(F9\"&dX$,,*&F(F/F8F4\"#f*&F8F4F(F4!#fF+\"#[F.!#IF (!#=,**$F8F9F4*$F5F9F4F@F4!\"\"F4,**&F8F4F5F4\"%NQF+!&%e>F.\"&()f#F(!% .k,**&F8F9F(F4FPF+\"%7pF.!%;pF(F>,.*$F8F/F6F7F6F8!%qwF:!&wD(F=\"'Lr5F@ !&dX$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "op(remove(has,GB2, \{x,y\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%\"zG\"\"(\"%_6*$F% \"\"&!%j<*$F%\"\"$\"$b'F%!#W" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sol_z:=solve(%,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol_zG6 )\"\"!\"\"\"#!\"#\"\"$#\"\"#F*!\"\",$*$\"#A#F'F,#F'\"#;,$F/#F-F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Substituting into the system yield s the possible tuples where the optima take place." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "map(op,[seq(map(`union`,[solve(convert(subs (z=i,GB2),set),\{x,y\})],\{z=i\}),i=sol_z)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,<%/%\"zG\"\"!/%\"xGF'/%\"yG\"\"\"<%F%F(/F+!\"\"<%F%/F +F'/F)F,<%F%F1/F)F/<%F(F1/F&F,<%/F&#!\"#\"\"$/F)F9/F+#F/F;<%/F&#\"\"#F ;F " 0 "" {MPLTEXT 1 0 50 "min(op(map(subs,%,phi))),max(op(map(subs, %,phi)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$#!#G\"#F\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Similar calculations using " } {XPPEDIT 18 0 "plex" "I%plexG6\"" }{TEXT -1 24 " need more than an hou r." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 190 "As another example, the Gr oebner package allows for the computation of standard bases. We defin e the leading term of a polynomial as its minimum term instead of its \+ maximum term by using a " }{XPPEDIT 18 0 "plex_min" "I)plex_minG6\"" } {TEXT -1 26 " term order rather than a " }{XPPEDIT 18 0 "plex" "I%plex G6\"" }{TEXT -1 12 " term order:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L:=1+x+y+z+x^2+y^2+z^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,0\"\"\"F&%\"xGF&%\"yGF&%\"zGF&*$F'\"\"#F&*$F(F+F&*$F)F+F& " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "leadterm(L,plex(x,y,z)) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$%\"xG\"\"#" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 28 "leadterm(L,plex_min(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The n Groebner bases can be computed for homogeneous ideals. Here is an e xample in " }{XPPEDIT 18 0 "Q*[x,y,z,w]" "*&%\"QG\"\"\"7&%\"xG%\"yG%\" zG%\"wGF$" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "G:=[y*w^2-z^3,x*w^3-z^4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"G G7$,&*&%\"yG\"\"\"%\"wG\"\"#F)*$%\"zG\"\"$!\"\",&*&%\"xGF)F*F.F)*$F-\" \"%F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Compare the usual Groebn er basis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "gbasis(G,plex(w ,x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7',&*&%\"yG\"\"$%\"zG\"\" &!\"\"*&F(\"\"'%\"xG\"\"#\"\"\",&*&F(F,F-F/F**(F&F.%\"wGF/F(\"\"%F/,&* (F3F/F-F/F(F'F/*&F&F/F(F4F*,&*&F&F/F3F.F/*$F(F'F*,&*&F-F/F3F'F/*$F(F4F *" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "which allows for divisions b y decreasing powers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "red uce(w^5*x^3*y^2*z,%,plex(w,x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&%\"zG\"\"(%\"yG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "and th e standard basis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "gbasis( G,plex_min(w,x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&,&*&%\"yG\" \"\"%\"wG\"\"#!\"\"*$%\"zG\"\"$F',&*&%\"xGF'F(F-F**(F,F'F&F'F(F)F',&*( F,F)F0F'F(F-F'*&F&F)F(\"\"%F*,&*(F(F5F0F)F,F'F**&F&F-F(F5F'" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "which allows for divisions by incr easing powers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "reduce(w^ 5*x^3*y^2*z,%,plex_min(w,x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#* (%\"wG\"\"'%\"xG\"\"%%\"yG\"\"\"" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 16 "Ideal invariants" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 275 "The di mension of a variety (zero for isolated points, one for lines, two for planes) is the Hilbert dimension of the corresponding annihilating id eal. The Groebner package allows one to compute this Hilbert dimensio n, together with the Hilbert polynomial and Hilbert series." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "As a first example, here is a zer o-dimensional ideal, corresponding to finitely many points in the vari ety." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "G:=[3*y^2-8*z^3,x^2 -2*z*x+5,3*y^3+8*x*y^2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "hilbertdim(G,tdeg(y,x,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "hilbertseries(G,tdeg(y,x, z),s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$%\"sG\"\"&\"\"\"*$F%\"\" %\"\"$*$F%F*F&*$F%\"\"#F&F%F*F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "subs(s=1,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#= " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "hilbertpoly(G,tdeg(y,x, z),s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "As another example, here is an ideal of positive dim ension." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "G:=[x^3*y^2+3*x^ 2*y^2+y^3+1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "hilbertdim (G,tdeg(x,y));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "hilbertseries(G,tdeg(x,y),s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,,*$%\"sG\"\"%\"\"\"*$F'\"\"$F)*$F '\"\"#F)F'F)F)F)F),&!\"\"F)F'F)F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "series(%,s,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+9 %\"sG\"\"\"\"\"!\"\"#\"\"\"\"\"$\"\"#\"\"%\"\"$\"\"&\"\"%F-\"\"&F-\"\" 'F-\"\"(F-\"\")F-\"\"*-%\"OG6#F%\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "hilbertpoly(G,tdeg(y,x),s);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "FGLM al gorithm" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "The FGLM algorithm was \+ introduced to perform changes of orderings for zero-dimensional ideals ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "G:=[x+y+z,x*y+y*z+z*x, x*y*z-1]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "GB:=gbasis(G,t deg(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7%,(%\"xG\"\"\"% \"yGF(%\"zGF(,(*$F)\"\"#F(*&F*F(F)F(F(*$F*F-F(,&!\"\"F(*$F*\"\"$F(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "hilbertdim(GB,tdeg(x,y,z)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "The implementation is a fully user-customizable generaliz ation:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "1. choice of the normal form:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "NFproc:=proc(t,NF ,T) local r,s; global GB;\n r:=reduce(t,GB,tdeg(x,y,z),s);\n NF[ t]:=r/s\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "2. computation o f dependencies between those normal forms:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 591 "FDproc:=proc(M,NF)\n local ind_lst,t,eta,sol,ze ro,sys,rel;\n ind_lst:=map(op,[indices(NF)]);\n for t in M do\n \+ ind_lst:=remove(divide,ind_lst,t)\n od;\n zero:=expand(nu mer(normal(add(eta[t]*NF[t],\n t=ind_lst))));\n sys:=\{coeff s(zero,\{x,y,z\})\};\n sol:=[readlib(`solve/linear`)(sys,\n \+ \{seq(eta[t],t=ind_lst)\})];\n rel:=subs(sol[1],add(eta[t]*t,t=ind _lst));\n `if`(rel=0,FAIL,\n collect(primpart(numer(subs(map (n->n=1,\n map2(op,1,select(evalb,sol[1]))),\n \+ rel)),\{x,y,z\}),\{x,y,z\},\n distributed,factor ))\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "3. choice of the term ination condition:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "TMpro c:=proc(border,monoideal,TOrd)\n border<>[]\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "A:=poly_algebra(x,y,z):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "T:=termorder(A,plex(z,y,x)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "FGLM:=[fglm(NFproc,FDproc,TMproc,T) ]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "GB2:=map(op,[entries( FGLM[2])]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$GB2G7%,&*$%\"xG\"\"$ !\"\"\"\"\"F+,(*$F(\"\"#F+*$%\"yGF.F+*&F(F+F0F+F+,(F(F+F0F+%\"zGF+" }} }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "Groebner bases for skew algebra s" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 264 "The functionality of the Gro ebner package is also available for skew polynomial inputs. Whereas i n the commutative case the ideals are two-sided, in the skew case we f ocus on left-sided ideal. This corresponds to skew polynomials acting on functions on the left." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "As \+ a first application, we derive a proof that " }{HYPERLNK 17 "Legendre \+ polynomials" 2 "orthopoly[P]" "" }{TEXT -1 8 " satisfy" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "Sum(orthopoly[P](n,z)*u^n,n=0..infi nity)=1/sqrt(1-2*z*u+u^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG 6$*&-&%*orthopolyG6#%\"PG6$%\"nG%\"zG\"\"\")%\"uGF.F0/F.;\"\"!%)infini tyG*$,(F0F0*&F/F0F2F0!\"#*$F2\"\"#F0#!\"\"F<" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 21 "Legendre polynomials " }{XPPEDIT 18 0 "orthopoly[P]" "& %*orthopolyG6#%\"PG" }{TEXT -1 66 " are defined by the following set o f operators in the Ore algebra " }{XPPEDIT 18 0 "C(z,n)*[Dz,Sn]" "*&-% \"CG6$%\"zG%\"nG\"\"\"7$%#DzG%#SnGF(" }{TEXT -1 36 " built over a diff erential operator " }{XPPEDIT 18 0 "Dz" "I#DzG6\"" }{TEXT -1 22 " and \+ a shift operator " }{XPPEDIT 18 0 "Sn" "I#SnG6\"" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "G:=\{(1-z^2)*Dz^2-2*z*Dz+n* (n+1),\n (n+2)*Sn^2-(2*n+3)*z*Sn+(n+1),\n (1-z^2)*Dz*Sn+(n+1)*z* Sn-(n+1)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GG<%,**&,&%\"nG\"\" \"\"\"#F*F*%#SnGF+F**(,&F)F+\"\"$F*F*%\"zGF*F,F*!\"\"F)F*F*F*,**(,&F*F **$F0F+F1F*%#DzGF*F,F*F**(,&F)F*F*F*F*F0F*F,F*F*F)F1F1F*,(*&F4F*F6F+F* *&F0F*F6F*!\"#*&F)F*F8F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "It \+ follows that the summand " }{XPPEDIT 18 0 "orthopoly[P](n,z)*u^n" "*&- &%*orthopolyG6#%\"PG6$%\"nG%\"zG\"\"\")%\"uGF)F+" }{TEXT -1 43 " in th e identity we want to prove satisfies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "G:=map(numer@normal,subs(Sn=Sn/u,G)) union \{u*Du-n\} ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GG<&,&*&%\"uG\"\"\"%#DuGF)F)% \"nG!\"\",.*&%#SnG\"\"#F+F)F)*$F/F0F0**%\"zGF)F/F)F(F)F+F)!\"#*(F3F)F/ F)F(F)!\"$*&F+F)F(F0F)*$F(F0F),,*$%#DzGF0F)*&F;F0F3F0F,*&F3F)F;F)F4*$F +F0F)F+F),.*&F;F)F/F)F)*(F;F)F/F)F3F0F,*(F3F)F/F)F+F)F)*&F3F)F/F)F)*&F +F)F(F)F,F(F," }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "in the new algeb ra " }{XPPEDIT 18 0 "C(z,u)*[n]*[Dz,Du,Sn]" "*(-%\"CG6$%\"zG%\"uG\"\" \"7#%\"nGF(7%%#DzG%#DuG%#SnGF(" }{TEXT -1 36 ". (Since the index of s ummation is " }{XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 73 ", we conside r polynomials in this variable to allow for its elimination.)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "A:=skew_algebra(diff=[Dz,z], diff=[Du,u],shift=[Sn,n],polynom=n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "T:=termorder(A,lexdeg([n],[Dz,Du,Sn])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "GB:=gbasis(G,T);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%#GBG7',,%\"uG\"\"\"**%\"zGF(%#SnGF(F'F(%#DuGF(!\"\" *&%#DzGF(F+F(F-*(F/F(F+F(F*\"\"#F(*&F'F1F,F(F(,,*&F*F(F/F(F1*$F/F1F-*& F/F1F*F1F(*&F'F1F,F1F-*&F'F(F,F(!\"#,,F'F(*&F*F(F+F(F-F)F9F2F(*&F,F(F+ F1F(,4*(F/F(F'F(F,F(F(**F*F1F/F(F'F(F,F(F-*(F+F(F'F(F,F1F(*&F/F(F*F1F- F/F(*&F,F(F+F(F(F*F-*(F*F(F'F(F,F(!\"$*(F*F(F'F1F,F1F-,&F8F-%\"nGF(" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "The previous equations are satis fied by the summand. Setting " }{XPPEDIT 18 0 "Sn=1" "/%#SnG\"\"\"" } {TEXT -1 107 " in them yields equations satisfied by the sum. A final Groebner basis calculation returns these equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "GB:=gbasis(subs(Sn=1,remove(has,GB,n)),T) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "collect(GB,\{Dz,Du\},d istributed,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$,(%\"uG!\"\"% \"zG\"\"\"*&,(*&F'F(F%F(\"\"#*$F%F,F&F&F(F(%#DuGF(F(,&F%F(*&F*F(%#DzGF (F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Solving this simple system of PDE's yields the announced closed form." }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 89 "The same method applies to perform definite integration . Here, we show that the integral" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "Int((1+x*y+2*y^2)*exp(x^2*y^2/(1+2*y^2))/y^n/y/(1+2*y ^2)^(2/3),\n y=-infinity..infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*,,(\"\"\"F(*&%\"xGF(%\"yGF(F(*$F+\"\"#F-F(-%$expG6#*(F *F-F+F-,&F(F(F,F-!\"\"F()F+%\"nGF3F+F3F2#!\"#\"\"$/F+;,$%)infinityGF3F <" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 284 "satisfies a recurrence of o rder 6. With a more refined method also based on Groebner basis calcu lations, one could obtain a second order recurrence so as to get a pro of that the integral is an integral representation for Hermite polynom ials. These polynomials are known to Maple as " }{XPPEDIT 18 0 "ortho poly[H](n,x)" "-&%*orthopolyG6#%\"HG6$%\"nG%\"xG" }{TEXT -1 1 "." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "They satisfy the system" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "G:=\{Sn*y-1,\n 3*(1+x*y+ 2*y^2)*(1+2*y^2)^2*y*Dy+24*n*y^6+8*y^6+12*n*x*y^5+16*x*y^5\n\011-12*x^ 2*y^4+20*y^4+36*n*y^4+12*n*x*y^3+8*x*y^3-6*x^3*y^3+14*y^2\n\011-6*x^2* y^2+18*n*y^2+3*n*x*y+3+3*n\};" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"G G<$,&*&%#SnG\"\"\"%\"yGF)F)!\"\"F),D**,(F)F)*&%\"xGF)F*F)F)*$F*\"\"#F2 F),&F)F)F1F2F2F*F)%#DyGF)\"\"$*&%\"nGF)F*\"\"'\"#C*$F*F8\"\")*(F7F)F0F )F*\"\"&\"#7*&F0F)F*F=\"#;*&F0F2F*\"\"%!#7*$F*FB\"#?*&F7F)F*FB\"#O*(F7 F)F0F)F*F5F>*&F*F5F0F)F;*&F0F5F*F5!\"'F1\"#9*&F0F2F*F2FK*&F7F)F*F2\"#= *(F7F)F0F)F*F)F5F5F)F7F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "in th e Ore algebra " }{XPPEDIT 18 0 "C(x,n)*[y]*[Dy,Sn]" "*(-%\"CG6$%\"xG% \"nG\"\"\"7#%\"yGF(7$%#DyG%#SnGF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "A:=skew_algebra(diff=[Dy,y],shift=[Sn,n],comm=x,polyn om=y):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "built on a differential operator " }{XPPEDIT 18 0 "Dy" "I#DyG6\"" }{TEXT -1 22 " and a shift \+ operator " }{XPPEDIT 18 0 "Sn" "I#SnG6\"" }{TEXT -1 41 ". (Since the \+ variable of integration is " }{XPPEDIT 18 0 "y" "I\"yG6\"" }{TEXT -1 73 ", we consider polynomials in this variable to allow for its elimin ation.)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "The elimination is obt ained by using a suitable term order." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "T:=termorder(A,lexdeg([y],[Dy,Sn])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "GB:=gbasis(G,T);" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#>%#GBG7%,R*(%#SnG\"\"'%\"nG\"\"\"%\"xGF+\"\"$*&F(F)F, F+\"#:*&F(F-F,\"\"#!#7*$F(F-\"##**&F(F-F*F+\"#OF(\"\")*&F*F+F(F+\"#C*& F(\"\"(F*F+F-*$F(F;\"#@*&%#DyGF+F(F)F-*(F(\"\"&F,F+F?F+F-*&F?F+F(F1F6F ?F9*(F(F-F,F+F?F+\"#7*(F(\"\"%F*F+F,F+FD*&F(FFF,F-!\"'*&F(FFF,F+\"#W*& F(F1F,F+\"#G*(F(F1F*F+F,F+FD*(F(F+F,F+F?F+FD*&F?F+F(FF\"#=*$F(FA\"#')* &F(FAF*F+FP*&F(FAF,F1FH,&*&F(F+%\"yGF+F+!\"\"F+,R*&FWF+F?F+F9*&F?F+F(F AF-*&F(F)F*F+F-*$F(F)FP*(F(FFF,F+F?F+F-*(F(FAF*F+F,F+F-*&F(FAF,F+FD*&F ?F+F(F-FP*&F(FFF*F+FP*$F(FF\"#o*&F(FFF,F1FH*(F(F1F,F+F?F+FD*&F(F-F,F-F H*(F(F-F*F+F,F+FD*&F(F-F,F+\"#K*&F?F+F(F+F6*&F(F1F,F1F2*&F(F1F*F+F6*$F (F1\"#c*&F?F+F,F+FD*(F(F+F*F+F,F+FD*&F(F+F,F+\"#;F7F+F*F9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The above equations are also satisfied by the integrand. Setting " }{XPPEDIT 18 0 "Dy=0" "/%#DyG\"\"!" }{TEXT -1 54 " in them yields an equation satisfied by the integral." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "collect(primpart(subs(Dy=0,o p(remove(has,GB,y))),Sn),Sn,factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 " 6#,0*&,&%\"nG\"\"$\"#@\"\"\"F)%#SnG\"\"(F)*(F*\"\"',&F&F)\"\"&F)F)%\"x GF)F'*&,(\"#')F)F&\"#=*$F0\"\"#!\"'F)F*F/F)*(F0F),(F&F7!#AF)F5F'F)F*\" \"%!\"#*&,(\"##*F)F5!#7F&\"#OF)F*F'F)*(F0F),&F+F)F&F'F)F*F6F;*&,&\"\") F)F&\"#CF)F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "The order is \+ easily reduced by 1 to get the announced 6 order equation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "collect(subs(n=n-1,%/Sn),Sn,factor) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2*&,&\"#=\"\"\"%\"nG\"\"$F'%#SnG \"\"'F'*(,&F(F'\"\"%F'F'%\"xGF'F*\"\"&F)*&,(*$F/\"\"#!\"'F(F&\"#oF'F'F *F.F'*(F/F',(F(F5!#;F'F3F)F'F*F)!\"#*&,(F3!#7F(\"#O\"#cF'F'F*F4F'*(F/F ',&F.F'F(F)F'F*F'F.F9F'F(\"#C" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "Groebner bases for modules" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "Groebner bases for modules are available in the case of K[x[1],...,x[ p]]-modules of the form K[x[1],...,x[p],s[1],...,s[q]], where the " } {XPPEDIT 18 0 "x[i]" "&%\"xG6#%\"iG" }{TEXT -1 11 "'s and the " } {XPPEDIT 18 0 "s[i]" "&%\"sG6#%\"iG" }{TEXT -1 101 "'s need not commut e. Those module structures are obtained by disallowing the multiplica tions by the " }{XPPEDIT 18 0 "s[i]" "&%\"sG6#%\"iG" }{TEXT -1 26 "'s \+ along all calculations." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "As an \+ example of a Groebner basis for the " }{XPPEDIT 18 0 "K*[x,y,z]" "*&% \"KG\"\"\"7%%\"xG%\"yG%\"zGF$" }{TEXT -1 8 "-module " }{XPPEDIT 18 0 " K*[x,y,z,s]" "*&%\"KG\"\"\"7&%\"xG%\"yG%\"zG%\"sGF$" }{TEXT -1 44 ", w e simulaneously compute a Groebner basis " }{XPPEDIT 18 0 "GB" "I#GBG6 \"" }{TEXT -1 27 " for an ideal generated by " }{XPPEDIT 18 0 "G" "I\" GG6\"" }{TEXT -1 36 ", and we compute each polynomial in " }{XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 45 " as a combination of the initial pol ynomials." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The initial basis " }{XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 4 " is:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 31 "G:=[x+y+z,x*y+y*z+z*x,x*y*z-1];" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"GG7%,(%\"xG\"\"\"%\"yGF(%\"zGF(,(*&F'F(F)F(F (*&F*F(F)F(F(*&F'F(F*F(F(,&*(F'F(F)F(F*F(F(!\"\"F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Let us introduce the module generated by" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "MG:=[seq(s^(i-1)+G[i]*s^nops (G),i=1..nops(G))];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#MGG7%,&\"\" \"F'*&,(%\"xGF'%\"yGF'%\"zGF'F'%\"sG\"\"$F',&F-F'*&,(*&F*F'F+F'F'*&F,F 'F+F'F'*&F*F'F,F'F'F'F-F.F',&*$F-\"\"#F'*&,&*(F*F'F+F'F,F'F'!\"\"F'F'F -F.F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "in the polynomial algebr a" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "A:=poly_algebra(x,y,z, s):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "To refrain from any multip lication by " }{XPPEDIT 18 0 "s" "I\"sG6\"" }{TEXT -1 71 " in the calc ulations, we use a third argument in the call to termorder." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "T:=termorder(A,lexdeg([s],[x ,y,z]),[s]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "MGB:=collec t(gbasis(MG,T),s);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$MGBG7(,**&,(% \"xG\"\"\"%\"yGF*%\"zGF*F*%\"sGF*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/,,*&,(*$F+F4F*F1F**$F,F4F*F*F -F4F**&,(*&F+F4F,F*F/*&F,F4F+F*F/F/F*F*F-F*F**&F+F4F,F4F*F+F*F,F*,&F*F **&F(F*F-\"\"$F*,**&F8F*F-FBF*F-F/F+F*F,F*,**&,&F/F**$F,FBF*F*F-FBF**$ F-F4F**&F,F*F-F*F/F:F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Up to 0 's, the Groebner basis " }{XPPEDIT 18 0 "GB" "I#GBG6\"" }{TEXT -1 5 " \+ for " }{XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 3 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "GB:=map(coeff,MGB,s,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#GBG7(\"\"!F&F&,(%\"xG\"\"\"%\"yGF)%\"zGF),(*$F*\" \"#F)*&F+F)F*F)F)*$F+F.F),&!\"\"F)*$F+\"\"$F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "as checked by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "gbasis(G,tdeg(x,y,z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%,(% \"xG\"\"\"%\"yGF&%\"zGF&,(*$F'\"\"#F&*&F(F&F'F&F&*$F(F+F&,&!\"\"F&*$F( \"\"$F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "Furthermore, we get th e following relations between the elements of " }{XPPEDIT 18 0 "G" "I \"GG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 " seq(coeff(MGB[i],s,3)=subs(dummy='G',add(coeff(MGB[i],s,j)*dummy[j+1], j=0..2)),i=1..3);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6%/\"\"!,&*&,(*&%\" xG\"\"\"%\"yGF*!\"\"*&F)F*%\"zGF*F,*&F.F*F+F*F,F*&%\"GG6#F*F*F**&,(F)F *F+F*F.F*F*&F16#\"\"#F*F*/F$,&*&,&F*F**(F)F*F+F*F.F*F,F*F0F*F**&F4F*&F 16#\"\"$F*F*/F$,(*&,(*&F+F7F.F7F*F+F*F.F*F*F0F*F**&,(*&F+F7F.F*F,*&F.F 7F+F*F,F,F*F*F5F*F**&,(*$F+F7F*F/F**$F.F7F*F*F>F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "And we get the Groebner basis for " }{XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 29 " in terms of the elements of " } {XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "seq(coeff(MGB[i],s,3)=subs(dummy='G',add(coeff(MGB [i],s,j)*dummy[j+1],j=0..2)),i=4..6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/,(%\"xG\"\"\"%\"yGF&%\"zGF&&%\"GG6#F&/,(*$F'\"\"#F&*&F(F&F'F&F&*$F (F/F&,&*&,&F'F&F(F&F&F)F&F&&F*6#F/!\"\"/,&F7F&*$F(\"\"$F&,(*&F(F/F)F&F &*&F(F&F5F&F7&F*6#F;F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "or, in \+ matrix notation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "matrix (3,1,[seq(coeff(MGB[i],s,3),i=4..6)])=\n matrix(3,3,[seq(subsop(seq (j+1=coeff(MGB[i],s,j),j=0..2),[0,0,0]),\n\011i=4..6)])&*matrix(3,1,G) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'MATRIXG6#7%7#,(%\"xG\"\"\"%\" yGF+%\"zGF+7#,(*$F,\"\"#F+*&F-F+F,F+F+*$F-F1F+7#,&!\"\"F+*$F-\"\"$F+-% #&*G6$-F%6#7%7%F+\"\"!F@7%,&F,F+F-F+F6F@7%F3,$F-F6F+-F%6#7%F(7#,(*&F*F +F,F+F+F2F+*&F*F+F-F+F+7#,&*(F*F+F,F+F-F+F+F6F+" }}}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 2 1 1805 }