ApCoCoA-1:CharP.GBasisModSquares

This article is about a function from ApCoCoA-1.

CharP.GBasisModSquares

Computing a Groebner Basis of a given ideal intersected with x^2-x for all indeterminates x.

Syntax

CharP.GBasisModSquares(Ideal:IDEAL):LIST

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function returns the reduced Groebner basis for the given ideal intersected with the ideal generated by x^2-x for all indeterminates. If x^2-x for all indeterminates is in the ideal (e.g. the set of zeros is a subset of {0,1}^n) this method should produce the Groebner Basis much faster!

Please be aware, that this is much more efficient if the term ordering is Lex, DegLex or DegRevLex. Otherwise, first a DegRevLex Groebner Basis is computed and then transformed with the FGLM.FGLM-algorithm.

• @param Ideal An Ideal.

• @return The reduced Groebner Basis of the given ideal.

Example

Use R::=QQ[x,y,z];
I:=Ideal(x-y^2,x^2+xy,y^3);
GBasis(I);

[x^2 + xy, -y^2 + x, -xy]
-------------------------------
Use Z::=ZZ[x,y,z];
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong

-------------------------------
I:=Ideal(x-y^2,x^2+xy,y^3);
CharP.GBasisModSquares(I);
-- WARNING: Coeffs are not in a field
-- GBasis-related computations could fail to terminate or be wrong
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[y, x]
-------------------------------

See also

FGLM.FGLM

GBasis

Introduction to CoCoAServer

Introduction to Groebner Basis in CoCoA

Representation of finite fields