<Text-field style="Heading 1" layout="Heading 1">Loading the Code</Text-field>

Extra Stuff for PolynomialIdeals

This worksheet and the attached code collect a number of routines that, for various reasons, never made it into Maple 10 as user-accessible commands. They range from dirty hacks to complete high-level operations, which will undoubtedly find their way into a future release. A number of them are simply interfaces to undocumented internal routines. The only common thread is that all of these routines are useful in some way, especially to those working in this area. So enjoy the freebie.

If you would like to contribute feedback or suggestions, feel free to email me at rpearcea@cecm.sfu.ca. We are currently working on the next release of PolynomialIdeals, but I'll be happy to discuss algorithms and current developments. I occassionaly post code on my website: http://www.cecm.sfu.ca/~rpearcea. I am sorry to say, however, that I can not accept any code from third parties.

restart;

Along with this worksheet, you should have a file called "piextras.mpl". This file contains Maple source code, and you will need to load it into your Maple session using the read command. Once loaded, the commands remain available until you quit or otherwise restart Maple.

read "piextras.mpl";

with(PolynomialIdeals); # load PolynomialIdeals

Warning, the assigned name `<,>` now has a global binding

Warning, the protected name `subset` has been redefined and unprotected

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

with(PIextras); # load the extra commands

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

<Text-field style="Heading 1" layout="Heading 1">Factoring Systems of Equations</Text-field>

PolynomialIdeals includes commands for computing prime and primary decompositions. These commands factor systems of equations into irreducible components, either discarding or preserving multiple solutions. For example:

J := <x*y-x, x^2, y^3-y^2>;

Simplify(PrimeDecomposition(J));

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

Simplify(PrimaryDecomposition(J));

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

These algorithms are quite robust, especially for zero-dimensional ideals (whose solutions are points). However for general ideals we sometimes get bogged down trying to reduce the problem to the zero-dimensional case. In this situation, we can draw on other routines in Maple 10 to split the system before computing a full decomposition.

TriangularDecomposition

This command uses RegularChains:-Triangularize to split the system into triangular components. A Groebner basis for the original system is not computed, however a lexicographic basis is computed for each component of the output. The algorithm performs well on problems with high degree in a small number of variables, and is substantially faster if the input is a lexicographic Groebner basis.

rose := <7*y^4-20*x^2, 2160*x^2*z^4+1512*x*z^4+315*z^4-4000*x^2-2800*x-490, 40320000*x^6*y^2*z+67200000*x^5*y^3+28800000*x^5*y^2*z+94080000*x^4*y^3-23520000*x^4*y*z^2-10080000*x^4*z^3+40924800*x^3*y^3+21168000*x^3*y^2*z-41395200*x^3*y*z^2-28224000*x^3*z^3+2634240*x^2*y^3+4939200*x^2*y^2*z-26726560*x^2*y*z^2-15288000*x^2*z^3-2300844*x*y^3+347508*x*y^2*z-7727104*x*y*z^2-1978032*x*z^3-432180*y^3-852355*y*z^2-180075*z^3>:

P := TriangularDecomposition(rose):

seq(IsPrime(i),i=P);

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

IdealInfo:-KnownGroebnerBases(rose);

HeuristicDecomposition

This command uses the factoring Buchberger algorithm implemented in Groebner:-Solve. The command initially computes a lexicographic Groebner basis, and lexicographic bases are known for every component of the output. The algorithm performs well on problems with low degree with a high number of variables, but can not be used mod p.

butcher := <12*x*z*v^2+12*t*w^3-12*w^4+12*t*w^2-18*w^3+8*t*w-14*w^2-w-1, 8*x*z*u*v+8*t*w^3-8*w^4+4*t*w^2-12*w^3+4*t*w-14*w^2-3*w-1, -24*t*w^3+24*w^4-24*t*w^2+36*w^3-8*t*w+26*w^2+7*w+1, 2*z*u+2*y*v+2*t*w-2*w^2-w-1, 3*z*u^2+3*y*v^2-3*t*w^2+3*w^3+3*w^2-t+4*w, 4*z*u^3+4*y*v^3+4*t*w^3-4*w^4-6*w^3+4*t*w-10*w^2-w-1, 6*x*z*v-6*t*w^2+6*w^3-3*t*w+6*w^2-t+4*w>:

P := HeuristicDecomposition(butcher);

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

seq(IsPrime(i),i=P);

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

IdealInfo:-KnownGroebnerBases(butcher);

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

It is important to note that the commands above may not factor the system into prime components, for example:

J := <x^2-2, y^2-2>;

TriangularDecomposition(J);

HeuristicDecomposition(J);

Simplify(PrimeDecomposition(J));

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

EquidimensionalDecomposition

This command is not a heuristic, it factors systems of equations into components of distinct dimension. Systems of dimension zero correspond to points in affine space, while systems of dimension two or three correspond to curves or surfaces respectively. The dimension of an arbitrary system is the largest dimension among its components. This command separates the components of different dimension.

The system below is composed of five points, three curves, and a surface in three dimensional affine space.

J := Intersect(<x-1,y+1,z>, <x^2+y-z,z^3-x>, <y^2+x^3-x-z>);

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

E := EquidimensionalDecomposition(J);

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

seq(HilbertDimension(i),i=E);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNictSSNtbkdGJTY5USIwRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHRjgvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHUSV0cnVlRigvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkYvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGKC8lKW1hdGhzaXplR0Y1LUkjbW9HRiU2M1EiLEYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGTS8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR1EzdmVyeXRoaWNrbWF0aHNwYWNlRigvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKG1heHNpemVHUSlpbmZpbml0eUYoLyUobWluc2l6ZUdRIjFGKC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUwZm9udF9zdHlsZV9uYW1lR0ZYLyUlc2l6ZUdGNS8lK2ZvcmVncm91bmRHRkMvJStiYWNrZ3JvdW5kR0ZGLUYtNjlGaHBGMEYzRjZGOUY7Rj1GP0ZBRkRGR0ZJRktGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GXm8tRi02OVEiMkYoRjBGM0Y2RjlGO0Y9Rj9GQUZERkdGSUZLRk5GUEZSRlRGVkZZRmVuRmduRmluRlxvNyM2JSIiISIiIiIiIw==

P := PrimeDecomposition(J);

seq(HilbertDimension(i),i=P);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNjMtSSNtbkdGJTY5USIwRigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHRjgvJSp1bmRlcmxpbmVHRjgvJSpzdWJzY3JpcHRHRjgvJSxzdXBlcnNjcmlwdEdGOC8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHUSV0cnVlRigvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUqbWF0aGNvbG9yR0ZDLyUvbWF0aGJhY2tncm91bmRHRkYvJStmb250ZmFtaWx5R0YyLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGKC8lKW1hdGhzaXplR0Y1LUkjbW9HRiU2M1EiLEYoLyUlZm9ybUdRJmluZml4RigvJSZmZW5jZUdGOC8lKnNlcGFyYXRvckdGTS8lJ2xzcGFjZUdRJDBlbUYoLyUncnNwYWNlR1EzdmVyeXRoaWNrbWF0aHNwYWNlRigvJSlzdHJldGNoeUdGOC8lKnN5bW1ldHJpY0dGOC8lKG1heHNpemVHUSlpbmZpbml0eUYoLyUobWluc2l6ZUdRIjFGKC8lKGxhcmdlb3BHRjgvJS5tb3ZhYmxlbGltaXRzR0Y4LyUnYWNjZW50R0Y4LyUwZm9udF9zdHlsZV9uYW1lR0ZYLyUlc2l6ZUdGNS8lK2ZvcmVncm91bmRHRkMvJStiYWNrZ3JvdW5kR0ZGLUYtNjlGaHBGMEYzRjZGOUY7Rj1GP0ZBRkRGR0ZJRktGTkZQRlJGVEZWRllGZW5GZ25GaW5GXG9GXm9GZ3FGXm9GZ3FGXm9GLEZeb0YsRl5vLUYtNjlRIjJGKEYwRjNGNkY5RjtGPUY/RkFGREZHRklGS0ZORlBGUkZURlZGWUZlbkZnbkZpbkZcb0Zeb0YsRl5vRiw3IzYrIiIhIiIiRl9yRl9yRl5yRl5yIiIjRl5yRl5y

<Text-field style="Heading 1" layout="Heading 1">Finite Dimensional Vector Spaces</Text-field>

When I is zero-dimensional, the quotient ring k[x[1],...,x[n]]/I is a finite dimensional vector space over the monomials not reducible by I. The number of elements in a basis for this space corresponds to the number of solutions (counting multiplicities) of I over the algebraic closure of k.

NumberOfSolutions

This command computes the number of solutions over the algebraic closure of the coefficient field.

rose;

NumberOfSolutions(rose);

NiQtSSNtbkc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGKDY5USQxMzZGKC8lJ2ZhbWlseUdRMFRpbWVzfk5ld35Sb21hbkYoLyUlc2l6ZUdRIzEyRigvJSVib2xkR1EmZmFsc2VGKC8lJ2l0YWxpY0dGNS8lKnVuZGVybGluZUdGNS8lKnN1YnNjcmlwdEdGNS8lLHN1cGVyc2NyaXB0R0Y1LyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GKC8lJ29wYXF1ZUdGNS8lK2V4ZWN1dGFibGVHRjUvJSlyZWFkb25seUdRJXRydWVGKC8lKWNvbXBvc2VkR0Y1LyUqY29udmVydGVkR0Y1LyUraW1zZWxlY3RlZEdGNS8lLHBsYWNlaG9sZGVyR0Y1LyUwZm9udF9zdHlsZV9uYW1lR1EqMkR+T3V0cHV0RigvJSptYXRoY29sb3JHRkAvJS9tYXRoYmFja2dyb3VuZEdGQy8lK2ZvbnRmYW1pbHlHRi8vJSxtYXRodmFyaWFudEdRJ25vcm1hbEYoLyUpbWF0aHNpemVHRjI3IyIkTyI=

tord := IdealInfo:-KnownGroebnerBases(rose)[1];

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

MonomialBasis

This command computes a basis for k[x[1],...,x[n]]/I as a vector space. It returns a list of the monomials not reducible by a Groebner basis for I sorted in descending order.

ConvertToVector

This command writes a polynomial as a list of coefficients relative to a monomial basis. Any terms not present in the basis are truncated, so make sure that the polynomial is in normal form.

M := MonomialBasis(rose,tord);

6$-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF(6%-I#miGF%69Q"MF(/%'familyGQ0Times~New~RomanF(/%%sizeGQ#12F(/%%boldGQ&falseF(/%'italicGQ%trueF(/%*underlineGF8/%*subscriptGF8/%,superscriptGF8/%+foregroundGQ*[0,0,255]F(/%+backgroundGQ.[255,255,255]F(/%'opaqueGF8/%+executableGF8/%)readonlyGF;/%)composedGF8/%*convertedGF8/%+imselectedGF8/%,placeholderGF8/%0font_style_nameGQ*2D~OutputF(/%*mathcolorGFD/%/mathbackgroundGFG/%+fontfamilyGF2/%,mathvariantGQ'italicF(/%)mathsizeGF5-I#moGF%63Q#:=F(/%%formGQ&infixF(/%&fenceGF8/%*separatorGF8/%'lspaceGQ/thickmathspaceF(/%'rspaceGF[p/%)stretchyGF8/%*symmetricGF8/%(maxsizeGQ)infinityF(/%(minsizeGQ"1F(/%(largeopGF8/%.movablelimitsGF8/%'accentGF8/%0font_style_nameGFX/%%sizeGF5/%+foregroundGFD/%+backgroundGFG-F$6%-F_o63Q"[F(/FcoQ'prefixF(/FfoF;Fgo/FjoQ.thinmathspaceF(/F]pF_r/F_pF;F`pFbpFepFhpFjpF\qF^qF`qFbqFdq-F$6[\l-F$6%-I%msupGF%6%-F-69Q"yF(F0F3F6F9F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFinF\o-I#mnGF%69Q"3F(F0F3F6/F:F8F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgn/FjnQ'normalF(F\o/%1superscriptshiftGQ"0F(-F_o63Q1&InvisibleTimes;F(FboFeoFgo/FjoQ$0emF(/F]pFjsF^pF`pFbpFepFhpFjpF\qF^qF`qFbqFdq-Fgr6%-F-69Q"xF(F0F3F6F9F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFinF\o-F]s69Q"7F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcs-F_o63Q",F(FboFeo/FhoF;Fis/F]pQ3verythickmathspaceF(F^pF`pFbpFepFhpFjpF\qF^qF`qFbqFdq-F$6#-Fgr6%-F-69Q"zF(F0F3F6F9F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFinF\o-F]s69Q"9F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcsFdt-F$6%FirFfs-Fgr6%F^u-F]s69Q"8F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcsFdt-F$6%-Fgr6%F^u-F]s69Q"6F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcsFfsFfrFdt-F$6'FfrFfs-Fgr6%F^tF\sFcsFfs-Fgr6%F^uF\sFcsFdt-F$6'FfrFfs-Fgr6%F^t-F]s69Q"4F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcsFfs-Fgr6%F^u-F]s69Q"2F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcsFdt-F$6'-Fgr6%F^t-F]s69Q"5F(F0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\oFcsFfsFfvFfsFirFdt-F$6'FfwFfs-Fgr6%FirFawFcsFfsF_wFdt-F$6'FfwFfsF^uFfsFfrFdt-F$6%FfvFfs-Fgr6%F^tF_vFcsFdt-F$6'FcxFfsFirFfsF_wFdt-F$6%FfrFfsFcxFdt-F$6%F\tFfsF_wFdt-F$6'FirFfsF\tFfsF^uFdt-F$6%F\tFfsF]xFdt-F$6%F^uFfs-Fgr6%F^tFhuFcsFdt-F$6%FirFfsFayFdt-F$6#-Fgr6%F^tFauFcsFdt-F$6#FfuFdt-F$6%-Fgr6%F^uFatFcsFfsFirFdt-F$6%F]vFfsF]xFdt-F$6%-Fgr6%F^uFhwFcsFfsFfrFdt-F$6'FirFfsF]vFfsF^tFdt-F$6'-Fgr6%F^uF\wFcsFfsF^tFfsFfrFdt-F$6'FfrFfs-Fgr6%F^tFawFcsFfsFfvFdt-F$6'F]xFfsFdvFfsFfvFdt-F$6'FfrFfsFdvFfsF_wFdt-F$6'FjvFfsFfvFfsFirFdt-F$6'F]xFfsFjvFfsF_wFdt-F$6'FjvFfsF^uFfsFfrFdt-F$6%FfwFfsFfvFdt-F$6'FfwFfsFirFfsF_wFdt-F$6'FfwFfsF]xFfsF^uFdt-F$6%FfwFfsFfrFdt-F$6%F_wFfsFcxFdt-F$6'F^uFfsFirFfsFcxFdt-F$6%F]xFfsFcxFdt-F$6%F\tFfsF^uFdt-F$6%FirFfsF\tFdt-F$6#FayFdt-F$6#F]zFdt-F$6%FirFfsF]vFdt-F$6%FczFfsF]xFdt-F$6%FfrFfsFizFdt-F$6%F]vFfsF^tFdt-F$6'FczFfsFirFfsF^tFdt-F$6'F^tFfsFizFfsF]xFdt-F$6'F^tFfsFfvFfsFfrFdt-F$6'FfvFfsF]xFfsF][lFdt-F$6'FfrFfsF][lFfsF_wFdt-F$6'FfvFfsFirFfsFdvFdt-F$6'FdvFfsF_wFfsF]xFdt-F$6'FdvFfsF^uFfsFfrFdt-F$6%FjvFfsFfvFdt-F$6'FjvFfsFirFfsF_wFdt-F$6'F]xFfsF^uFfsFjvFdt-F$6%FjvFfsFfrFdt-F$6%F_wFfsFfwFdt-F$6'FfwFfsFirFfsF^uFdt-F$6%FfwFfsF]xFdt-F$6%FcxFfsF^uFdt-F$6%FirFfsFcxFdt-F$6#F\tFdt-F$6#F]vFdt-F$6%FirFfsFczFdt-F$6%FizFfsF]xFdt-F$6%FfvFfsFfrFdt-F$6%FczFfsF^tFdt-F$6'FirFfsFizFfsF^tFdt-F$6'F^tFfsFfvFfsF]xFdt-F$6'FfrFfsF^tFfsF_wFdt-F$6'F][lFfsFfvFfsFirFdt-F$6'F][lFfsF_wFfsF]xFdt-F$6'F][lFfsF^uFfsFfrFdt-F$6%FdvFfsFfvFdt-F$6'FdvFfsFirFfsF_wFdt-F$6'FdvFfsF]xFfsF^uFdt-F$6%FdvFfsFfrFdt-F$6%F_wFfsFjvFdt-F$6'F^uFfsFirFfsFjvFdt-F$6%F]xFfsFjvFdt-F$6%F^uFfsFfwFdt-F$6%FirFfsFfwFdt-F$6#FcxFdt-F$6#FczFdt-F$6%FizFfsFirFdt-F$6%FfvFfsF]xFdt-F$6%F_wFfsFfrFdt-F$6%F^tFfsFizFdt-F$6'FfvFfsF^tFfsFirFdt-F$6'F]xFfsF^tFfsF_wFdt-F$6'F^tFfsF^uFfsFfrFdt-F$6%F][lFfsFfvFdt-F$6'F][lFfsFirFfsF_wFdt-F$6'F][lFfsF]xFfsF^uFdt-F$6%F][lFfsFfrFdt-F$6%F_wFfsFdvFdt-F$6'F^uFfsFirFfsFdvFdt-F$6%F]xFfsFdvFdt-F$6%F^uFfsFjvFdt-F$6%FirFfsFjvFdt-F$6#FfwFdt-F$6#FizFdt-F$6%FirFfsFfvFdt-F$6%F_wFfsF]xFdt-F$6%F^uFfsFfrFdt-F$6%F^tFfsFfvFdt-F$6'F^tFfsFirFfsF_wFdt-F$6'F^tFfsF]xFfsF^uFdt-F$6%F^tFfsFfrFdt-F$6%F][lFfsF_wFdt-F$6'FirFfsF][lFfsF^uFdt-F$6%F]xFfsF][lFdt-F$6%F^uFfsFdvFdt-F$6%FirFfsFdvFdt-F$6#FjvFdt-F$6#FfvFdt-F$6%FirFfsF_wFdt-F$6%F^uFfsF]xFdt-F$6#FfrFdt-F$6%F^tFfsF_wFdt-F$6'FirFfsF^tFfsF^uFdt-F$6%F]xFfsF^tFdt-F$6%F][lFfsF^uFdt-F$6%FirFfsF][lFdt-F$6#FdvFdt-F$6#F_wFdt-F$6%FirFfsF^uFdt-F$6#F]xFdt-F$6%F^tFfsF^uFdt-F$6%F^tFfsFirFdt-F$6#F][lFdtF^uFdtFirFdtF^tFdt-F]s69FgpF0F3F6F`sF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFasF\o-F_o63Q"]F(/FcoQ(postfixF(F]rFgoF^r/F]pQ2verythinmathspaceF(FarF`pFbpFepFhpFjpF\qF^qF`qFbqFdq7#-_F)I,mprintslashGF(6$7#>I"MGF(7ds*&)I"yGF(""$""")I"xGF(""(F[jl*$)I"zGF(""*F[jl*&FiilF[jl)Fajl"")F[jl*&)Fajl""'F[jlFhilF[jl*(FhilF[jl)F]jlFjilF[jl)FajlFjilF[jl*(FhilF[jl)F]jl""%F[jl)Fajl""#F[jl*()F]jl""&F[jlF[[mF[jlFiilF[jl*(Fb[mF[jl)FiilF`[mF[jlF_[mF[jl*(Fb[mF[jlFajlF[jlFhilF[jl*&F[[mF[jl)F]jlFhjlF[jl*(Fh[mF[jlFiilF[jlF_[mF[jl*&FhilF[jlFh[mF[jl*&F\jlF[jlF_[mF[jl*(FiilF[jlF\jlF[jlFajlF[jl*&F\jlF[jlFe[mF[jl*&FajlF[jl)F]jlFejlF[jl*&FiilF[jlF_\mF[jl*$)F]jlFbjlF[jl*$FdjlF[jl*&)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f := -56*x*y^5*z+16*x^3*y^5*z-4*x^3*y^4*z^2+10*z^6*x^3+95*x^3*y^6*z+25*z^7*x^3;

f := NormalForm(f, rose, tord);

6$-I%mrowG6#/I+modulenameG6"I,TypesettingGI(_syslibGF(6%-I#miGF%69Q"fF(/%'familyGQ0Times~New~RomanF(/%%sizeGQ#12F(/%%boldGQ&falseF(/%'italicGQ%trueF(/%*underlineGF8/%*subscriptGF8/%,superscriptGF8/%+foregroundGQ*[0,0,255]F(/%+backgroundGQ.[255,255,255]F(/%'opaqueGF8/%+executableGF8/%)readonlyGF;/%)composedGF8/%*convertedGF8/%+imselectedGF8/%,placeholderGF8/%0font_style_nameGQ*2D~OutputF(/%*mathcolorGFD/%/mathbackgroundGFG/%+fontfamilyGF2/%,mathvariantGQ'italicF(/%)mathsizeGF5-I#moGF%63Q#:=F(/%%formGQ&infixF(/%&fenceGF8/%*separatorGF8/%'lspaceGQ/thickmathspaceF(/%'rspaceGF[p/%)stretchyGF8/%*symmetricGF8/%(maxsizeGQ)infinityF(/%(minsizeGQ"1F(/%(largeopGF8/%.movablelimitsGF8/%'accentGF8/%0font_style_nameGFX/%%sizeGF5/%+foregroundGFD/%+backgroundGFG-F$6in-F$6%-I&mfracGF%6*-I#mnGF%69Q-447917169665F(F0F3F6/F:F8F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgn/FjnQ'normalF(F\o-F^r69Q,16991987382F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o/%.linethicknessGQ"1F(/%+denomalignGQ'centerF(/%)numalignGF\s/%)bevelledGF8FbqFdq-F_o63Q1&InvisibleTimes;F(FboFeoFgo/FjoQ$0emF(/F]pFesF^pF`pFbpFepFhpFjpF\qF^qF`qFbqFdq-F$6'-I%msupGF%6%-F-69Q"xF(F0F3F6F9F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFinF\o-F^r69Q"2F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o/%1superscriptshiftGQ"0F(Fas-F-69Q"yF(F0F3F6F9F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFinF\oFas-Fjs6%-F-69Q"zF(F0F3F6F9F<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFinF\oF_tFbt-F_o63Q(&minus;F(FboFeoFgo/FjoQ0mediummathspaceF(/F]pFauF^pF`pFbpFepFhpFjpF\qF^qF`qFbqFdq-F$6%-F[r6*-F^r69Q.1255636272775F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q,67967949528F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6'FisFas-Fjs6%FetF_tFbtFasFjt-F_o63Q"+F(FboFeoFgoF`uFbuF^pF`pFbpFepFhpFjpF\qF^qF`qFbqFdq-F$6%-F[r6*-F^r69Q-325526139505F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q-135935899056F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6'F\tFasF_vFasFjtFav-F$6%-F[r6*-F^r69Q,52876596500F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q+2831997897F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6'-Fjs6%F\t-F^r69Q"3F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFbtFasFetFasFhtFav-F$6%-F[r6*-F^r69Q,98579171131F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q+8495993691F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6'F\tFasFetFasFhtFav-F$6%-F[r6*-F^r69Q-333106544701F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q-339839747640F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6#-Fjs6%FetF^xFbtFav-F$6%-F[r6*-F^r69Q-223585496533F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFjuFgrFjrF]sF_sFbqFdqFas-F$6%F\tFasFiyF]u-F$6%-F[r6*-F^r69Q,99460398730F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFhxFgrFjrF]sF_sFbqFdqFas-F$6%FisFasFiyFav-F$6%-F[r6*-F^r69Q.1582092282100F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q+6607995093F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6'-Fjs6%F\t-F^r69Q"5F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFbtFasF_vFasFjtFav-F$6%-F[r6*-F^r69Q,13670005000F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgwFgrFjrF]sF_sFbqFdqFas-F$6'F\xFasF_vFasFjtFav-F$6%-F[r6*-F^r69Q$500F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q#27F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6%FhtFasF\xFav-F$6%-F[r6*-F^r69Q-136632191087F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q-100693258560F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6#-Fjs6%FjtF^xFbtF]u-F$6%-F[r6*-F^r69Q-151075990000F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgwFgrFjrF]sF_sFbqFdqFas-F$6%-Fjs6%F\t-F^r69Q"4F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFbtFasFiyF]u-F$6%-F[r6*-F^r69Q-476719601750F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFhxFgrFjrF]sF_sFbqFdqFas-F$6%F\xFasFiyFav-F$6%-F[r6*-F^r69Q,51257470750F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q*943999299F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6%F\xFasF_^lFav-F$6%-F[r6*-F^r69Q-111123875975F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q+5663995794F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6%FisFasF_^lFav-F$6%-F[r6*-F^r69Q+1270754450F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q*314666433F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6%F\tFasF_^lFav-F$6%-F[r6*-F^r69Q.2219045676589F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q.1359358990560F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6%FetFasFhtFav-F$6%-F[r6*-F^r69Q#49F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r69Q#48F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFgrFjrF]sF_sFbqFdqFas-F$6#-Fjs6%Fjt-F^r69Q"6F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\oFbtF]u-F$6%-F[r6*-F^r69Q*883677515F(F0F3F6FarF<F>F@FBFEFHFJFLFNFPFRFTFVFYFenFgnFbrF\o-F^r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C := ConvertToVector(f,M,tord);

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f - add(C[i]*M[i], i=1..nops(C));

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

<Text-field style="Heading 1" layout="Heading 1">Vanishing Ideals and Multivariate Interpolation</Text-field>

The VanishingIdeal command computes a Groebner basis for an ideal whose solutions are a given set of points. For example:

L := [[1,2],[1,3],[0,5]];

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

J := VanishingIdeal(L,[x,y],plex(x,y));

Simplify(PrimeDecomposition(J),plex(x,y));

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The "points" can also be algebraic numbers or functions of a parameter.

L2 := [[1,2],[1,sqrt(2)],[a,0]];

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K := VanishingIdeal(L2,[x,y],tdeg(x,y));

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Simplify(PrimeDecomposition(K),tdeg(x,y));

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We demonstrate how to use this VanishingIdeal to do multivariate polynomial interpolation. The problem is to compute a polynomial f in k[x1,...,xn] passing through a given set of n+1 dimensional points. Since f is a function we can assume that the points contain distinct values in {x1,...,xn}. If Z is a new variable then a multiple of Z-f will be the largest element in a Groebner basis for the vanishing ideal if Z is greater than all of the other variables. In the example below, this is the polynomial Z-(x+y-2).

vals := [1,2,3];

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points := zip((a,b)->[op(a),b], L, vals);

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

VanishingIdeal(points, [x,y,Z], lexdeg([Z],[x,y]));

MultivariateInterpolation

This command computes a multivariate interpolating polynomial. It was written by Jeff Farr, who also wrote VanishingIdeal. The initial arguments are either a list of n dimensional points and a corresponding list of values, or just a list of n+1 dimensional points. The remaining arguments are the list of n variables and a monomial order.

f := MultivariateInterpolation(L,vals,[x,y],tdeg(x,y));

MultivariateInterpolation(points,[x,y],tdeg(x,y));

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

seq(eval(f,{x=i[1],y=i[2]}), i=L);

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

L2;

g := MultivariateInterpolation(L2, [sqrt(3), a^2-1, 2], [x,y], tdeg(x,y));

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

[seq(eval(g,{x=i[1],y=i[2]}), i=L2)];

map(simplify,%);

<Text-field style="Heading 1" layout="Heading 1">Groebner basis commands</Text-field>

PolynomialIdeals contains its own Groebner basis commands which have additional functionality not available through the standard commands. Much of this functionality is designed to make programming with ideals and Groebner bases easier.

GroebnerBasis

This command computes a Groebner basis for an ideal or a list or set of generators. Its standard syntax is compatible with the short syntax of Groebner[Basis].

G := GroebnerBasis(J,grlex(x,y));

If the first argument is a list or set of generators then an optional last argument can specify the ring characteristic. Below we compute a lex Groebner basis for G mod 11.

GroebnerBasis(G,plex(x,y),11);

If the second argument is a name instead of a term order GroebnerBasis will choose the term order and assign it to the name. The functionality and syntax is identical to the IdealInfo:-DefaultMonomialOrder command, in that it will use a known Groebner basis (and avoid a Groebner basis computation) if possible.

IdealInfo:-KnownGroebnerBases(butcher);

G := GroebnerBasis(butcher,'tord'); tord;

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G := GroebnerBasis(butcher,'tord','tdeg'); tord;

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G := GroebnerBasis(butcher,'tord','grlex'); tord;

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IdealInfo:-KnownGroebnerBases(butcher);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiUtSSNtb0dGJTYzUSJ8ZnJGKC8lJWZvcm1HUSdwcmVmaXhGKC8lJmZlbmNlR1EldHJ1ZUYoLyUqc2VwYXJhdG9yR1EmZmFsc2VGKC8lJ2xzcGFjZUdRLnRoaW5tYXRoc3BhY2VGKC8lJ3JzcGFjZUdGOy8lKXN0cmV0Y2h5R0Y1LyUqc3ltbWV0cmljR0Y4LyUobWF4c2l6ZUdRKWluZmluaXR5RigvJShtaW5zaXplR1EiMUYoLyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJTBmb250X3N0eWxlX25hbWVHUSoyRH5PdXRwdXRGKC8lJXNpemVHUSMxMkYoLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYoLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV1GKC1GJDYnLUYkNiUtSSNtaUdGJTY5USZncmxleEYoLyUnZmFtaWx5R1EwVGltZXN+TmV3flJvbWFuRigvJSVzaXplR0ZTLyUlYm9sZEdGOC8lJ2l0YWxpY0dGNS8lKnVuZGVybGluZUdGOC8lKnN1YnNjcmlwdEdGOC8lLHN1cGVyc2NyaXB0R0Y4LyUrZm9yZWdyb3VuZEdGVi8lK2JhY2tncm91bmRHRlkvJSdvcGFxdWVHRjgvJStleGVjdXRhYmxlR0Y4LyUpcmVhZG9ubHlHRjUvJSljb21wb3NlZEdGOC8lKmNvbnZlcnRlZEdGOC8lK2ltc2VsZWN0ZWRHRjgvJSxwbGFjZWhvbGRlckdGOC8lMGZvbnRfc3R5bGVfbmFtZUdGUC8lKm1hdGhjb2xvckdGVi8lL21hdGhiYWNrZ3JvdW5kR0ZZLyUrZm9udGZhbWlseUdGXm8vJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YoLyUpbWF0aHNpemVHRlMtRi02M1EwJkFwcGx5RnVuY3Rpb247RigvRjFRJmluZml4RigvRjRGOEY2L0Y6USQwZW1GKC9GPUZhci9GP0Y4RkBGQkZFRkhGSkZMRk5GUUZURlctRiQ2JS1GLTYzUSIoRihGMEYzRjZGOUY8Rj5GQEZCRkVGSEZKRkxGTkZRRlRGVy1GJDYvLUZpbjY5USJ4RihGXG9GX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGW3FGXXFGX3FGYXFGY3FGZXFGaHEtRi02M1EiLEYoRl1yRl9yL0Y3RjVGYHIvRj1RM3Zlcnl0aGlja21hdGhzcGFjZUYoRmNyRkBGQkZFRkhGSkZMRk5GUUZURlctRmluNjlRInlGKEZcb0Zfb0Zhb0Zjb0Zlb0Znb0Zpb0ZbcEZdcEZfcEZhcEZjcEZlcEZncEZpcEZbcUZdcUZfcUZhcUZjcUZlcUZocUZecy1GaW42OVEiekYoRlxvRl9vRmFvRmNvRmVvRmdvRmlvRltwRl1wRl9wRmFwRmNwRmVwRmdwRmlwRltxRl1xRl9xRmFxRmNxRmVxRmhxRl5zLUZpbjY5USJ0RihGXG9GX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGW3FGXXFGX3FGYXFGY3FGZXFGaHFGXnMtRmluNjlRInVGKEZcb0Zfb0Zhb0Zjb0Zlb0Znb0Zpb0ZbcEZdcEZfcEZhcEZjcEZlcEZncEZpcEZbcUZdcUZfcUZhcUZjcUZlcUZocUZecy1GaW42OVEidkYoRlxvRl9vRmFvRmNvRmVvRmdvRmlvRltwRl1wRl9wRmFwRmNwRmVwRmdwRmlwRltxRl1xRl9xRmFxRmNxRmVxRmhxRl5zLUZpbjY5USJ3RihGXG9GX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGW3FGXXFGX3FGYXFGY3FGZXFGaHEtRi02M1EiKUYoL0YxUShwb3N0Zml4RihGM0Y2RjkvRj1RMnZlcnl0aGlubWF0aHNwYWNlRihGPkZARkJGRUZIRkpGTEZORlFGVEZXRl5zLUYkNiUtRmluNjlRJXBsZXhGKEZcb0Zfb0Zhb0Zjb0Zlb0Znb0Zpb0ZbcEZdcEZfcEZhcEZjcEZlcEZncEZpcEZbcUZdcUZfcUZhcUZjcUZlcUZocUZqcUZkckZecy1GJDYlLUZpbjY5USV0ZGVnRihGXG9GX29GYW9GY29GZW9GZ29GaW9GW3BGXXBGX3BGYXBGY3BGZXBGZ3BGaXBGW3FGXXFGX3FGYXFGY3FGZXFGaHFGanFGZHItRi02M1EifGhyRihGaXRGM0Y2RjlGW3VGPkZARkJGRUZIRkpGTEZORlFGVEZXNyM8JS1JJmdybGV4R0YoNilJInhHRihJInlHRihJInpHRihJInRHRihJInVHRihJInZHRihJIndHRigtSSVwbGV4R0YoRl52LUkldGRlZ0dGKEZedg==

An optional last argument after the name and term order type specifies what variables to use.

G := GroebnerBasis(butcher,'tord',{y,z,t,w}); tord;

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

G := GroebnerBasis(butcher,'tord','plex',{y,z,t,w}); tord;

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

This syntax makes programming with Groebner bases fairly simple. In the program below we will automatically avoid the computation of a lex basis if one is known.

eval(HeuristicDecomposition);

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

The GroebnerBasis command automatically uses the FGLM and Groebner Walk algorithms to compute lexicographic or elimination Groebner bases. The method option shown below allows you to control which algorithm is used. There is no way to specify the starting basis however; the command chooses a good one automatically.

trinks := <45*p+35*s-165*b-36, 35*p+40*z+25*t-27*s, -9*w+15*p*t+20*z*s, 99*w-11*s*b+3*b^2, w*p+2*z*t-11*b^3, 15*w+25*p*s+30*z-18*t-165*b^2>;

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

infolevel[':-GroebnerBasis'] := 2:

# use FGLM (also computes a starting basis) GroebnerBasis(trinks,plex(z,t,w,p,s,b),method=fglm):

-> Buchberger algorithm

total reductions : 23 total time (sec) .300 -> FGLM algorithm

total monomials : 16 total time (sec) .100

# use the Groebner Walk (starts from the plex basis computed above) GroebnerBasis(trinks,plex(t,w,z,s,p,b),method=walk):

-> Groebner Walk

total walk steps : 2 total time (sec) .110

# use the Buchberger algorithm (starting from the ideal generators) GroebnerBasis(trinks,lexdeg([z,t,w],[p,s,b]),method=buchberger):

-> Buchberger algorithm

total reductions : 3 total time (sec) 0.80e-1

infolevel[':-GroebnerBasis'] := 1:

GroebnerBasis can also compute a change of basis matrix using the extended Buchberger algorithm. This matrix describes how to write the Groebner basis in terms of the original generators. For example:

F := [seq(randpoly([x,y,z],terms=3,degree=2,coeffs=rand(0..3)), i=1..3)];

G, C := GroebnerBasis(F,'tord',method=extended);

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

Matrix(C) &* Vector(F) = Vector(G); # display the matrix/vector equation

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

# C, G, and F satisfy this equation for j=1..nops(G) # G[j] = add(C[j][i]*F[i], i=1..nops(F));

NormalForm

Our NormalForm command can optionally output a list of quotients used in the division. Combined with the method=extended option above, this allows us to write a polynomial in terms of an arbitrary set of ideal generators.

f := expand(add(randpoly([x,y,z],degree=2,coeffs=rand(-5..5))*G[i], i=1..nops(G)));

f in <F>;

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

NormalForm(f, G, tord, 'Q');

# f = add(Q[j]*G[j], j=1..nops(G)); # recall the equation for G[j] above

'f' = Vector(Q) &* Matrix(C) &* F;

Vector[row](Q).Matrix(C).Vector(F);

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expand(f-%);

# two ways to write f in terms of F # f = add(Q[j]*add(C[j][i]*F[i], i=1..nops(F)), j=1..nops(Q)); # f = add( add(Q[j]*C[j][i], j=1..nops(Q))*F[i], i=1..nops(F));

<Text-field style="Heading 1" layout="Heading 1">Polynomial Computations</Text-field>

We have developed a number of smaller routines which may be of interest to people working with multivariate polynomials. Some of them implement new functionality, while others are simply a little more general than Maple's builtin routines.d

LeadingMonomial, LeadingCoefficient, LeadingTerm

These commands are analogous to the ones in the Groebner package, however they support matrix term orders with rational entries.

f := 3*x^2+x*y^3+5;

M := [[1,1/3],[1/2,1/8]]: Matrix(M);

LeadingTerm(f, 'matrix'(M,[x,y]));

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LeadingMonomial(f, 'matrix'(M,[x,y]), 'lc'), lc;

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They also support negative entries in the matrix, although this is not really safe for Groebner basis computations. The example below isn't even a monomial order. The commands don't check if the order is valid, they will just assume it is and possibly run forever.

badM := [[-2,-1],[1,0]]; # use only if you know what you are doing

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LeadingTerm(f,'matrix'(badM,[x,y]));

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Typing 'matrix' every time gets pretty tedious, so you can just type "mat" instead to get a matrix order. The matrix can be a list of lists, a Maple matrix, or a Maple Matrix.

LeadingTerm(f,mat(M,[x,y]));

LeadingTerm(f,mat(Matrix(M),[x,y]));

MatrixOrder

Every monomial order is equivalent to a matrix order. You can compute a matrix for an order using the MatrixOrder command. The columns are ordered by sorting the variables in the monomial order.

MatrixOrder(plex(x,y,z));

MatrixOrder(tdeg(x,y,z));

MatrixOrder(lexdeg([x,y,z],[t,u,v]));

You can also reorder the columns to your liking. In the example below, mat(M,[z,x,y]) is equivalent to the input order.

M := MatrixOrder(tdeg(x,y,z),[z,x,y]);

InitialForm, WeightedDegree

These commands extract the maximal terms or compute the degree of a polynomial with respect to a vector of weights on the variables.

f := x^3+x^2*y+y^2;

WeightedDegree(f, [1,2], [x,y]);

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InitialForm(f, [1,2], [x,y]);

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

coeffs(f,[x,y],'M'): [M]; map(WeightedDegree, [M], [1,2], [x,y]);

Of course these commands support negative and rational weights as well.

WeightedDegree(f, [-1/2,-1], [x,y]);

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

InitialForm(f, [-1/2,-1], [x,y]);

NiQtSSVtcm93RzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliR0YoNiMtSSVtc3VwR0YlNiUtSSNtaUdGJTY5USJ4RigvJSdmYW1pbHlHUTBUaW1lc35OZXd+Um9tYW5GKC8lJXNpemVHUSMxMkYoLyUlYm9sZEdRJmZhbHNlRigvJSdpdGFsaWNHUSV0cnVlRigvJSp1bmRlcmxpbmVHRjsvJSpzdWJzY3JpcHRHRjsvJSxzdXBlcnNjcmlwdEdGOy8lK2ZvcmVncm91bmRHUSpbMCwwLDI1NV1GKC8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRigvJSdvcGFxdWVHRjsvJStleGVjdXRhYmxlR0Y7LyUpcmVhZG9ubHlHRj4vJSljb21wb3NlZEdGOy8lKmNvbnZlcnRlZEdGOy8lK2ltc2VsZWN0ZWRHRjsvJSxwbGFjZWhvbGRlckdGOy8lMGZvbnRfc3R5bGVfbmFtZUdRKjJEfk91dHB1dEYoLyUqbWF0aGNvbG9yR0ZHLyUvbWF0aGJhY2tncm91bmRHRkovJStmb250ZmFtaWx5R0Y1LyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGKC8lKW1hdGhzaXplR0Y4LUkjbW5HRiU2OVEiM0YoRjNGNkY5L0Y9RjtGP0ZBRkNGRUZIRktGTUZPRlFGU0ZVRldGWUZmbkZobkZqbi9GXW9RJ25vcm1hbEYoRl9vLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGKDcjKiQpSSJ4R0YoIiIkIiIi

coeffs(f,[x,y],'M'): [M]; map(WeightedDegree, [M], [-1/2,-1], [x,y]);

TrailingTerm, TrailingMonomial, TrailingCoefficient

The TrailingTerm command computes the smallest term of a polynomial with respect to a monomial order. The TrailingMonomial and TrailingCoefficient commands return this term's monomial or coefficient, respectively.

f := randpoly([x,y,z],terms=10,degree=20);

TrailingTerm(f,tdeg(x,y,z));

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

TrailingMonomial(f,tdeg(x,y,z),'tc'), tc;

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

TrailingCoefficient(f,tdeg(x,y,z));

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

<Text-field style="Heading 1" layout="Heading 1">Ideal Operations</Text-field>

HilbertSeries, HilbertPolynomial

These commands call the corresponding Groebner commands but they choose a monomial order automatically. A Groebner basis computation is avoided if possible.

butcher;

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HilbertSeries(butcher,s);

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

HilbertPolynomial(butcher,s);

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

Homogenize

This command homogenizes polynomials and ideals with respect to a new variable. Homogenizing an ideal in general requires a Groebner basis. This command will always use a tdeg basis because it is easy to construct a so-called "good extension" of a tdeg orders.

f := randpoly([x,y],terms=4,coeffs=rand(-10..10));

Homogenize(f,h);

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

Homogenize(f,x); # produces garbage

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

J := <x^3-y^2, x*y-1>;

Jh := Homogenize(J,h);

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

IdealInfo:-KnownGroebnerBases(Jh);

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

This is a property of good extensions.

GroebnerBasis(Jh,'tord'); tord;

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

Homogenize(GroebnerBasis(J, tord), h);

<Text-field style="Heading 1" layout="Heading 1">Misc.</Text-field>

MakeMonic

This command makes the leading term of a polynomial monic with respect to a given monomial order.

f := randpoly([x,y,z]);

LeadingTerm(f,tdeg(x,y,z));

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

MakeMonic(f,tdeg(x,y,z));

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

SuggestVariableOrder

This command tries to order of the variables for more efficient Groebner basis computations. It is used internally by IdealInfo:-DefaultMonomialOrder if a Groebner basis for the ideal is not known. The first argument can be an ideal or a list or set of generators. An optional second argument specifies what indeterminates are variables (in case there are parameters).

trinks;

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SuggestVariableOrder(trinks);

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

F := [seq(randpoly([x,y,z,t,u,v],terms=5),i=1..4)];

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SuggestVariableOrder(F);

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SuggestVariableOrder(F,[x,y,z,t]);

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

SuggestVariableOrder(F,{t,u,v,w});

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

LaTeX export

Finally Maple's latex command has been extended to properly handle PolynomialIdeals. For example:

J := <x^2-y, y^2+1>;

latex(J);

\langle {{y}^{2}+1,{x}^{2}-y}\rangle