D.4.34 symodstd_lib

Library:

symodstd.lib

Purpose:

Procedures for computing Groebner basis of ideals being invariant under certain variable permutations.

Author:

Stefan Steidel, [email protected]

Overview:

A library for computing the Groebner basis of an ideal in the polynomial ring over the rational numbers, that is invariant under certain permutations of the variables, using the symmetry and modular methods. More precisely let I = <f1,...,fr> be an ideal in Q[x(1),...,x(n)] and sigma a permutation of order k in Sym(n) such that sigma(I) = I. We assume that sigma({f1,...,fr}) = {f1,...,fr}. This can always be obtained by adding sigma(fi) to {f1,...,fr}.
To compute a standard basis of I we apply a modification of the modular version of the standard basis algorithm (improving the calculations in positive characteristic). Therefore we only allow primes p such that p-1 is divisible by k. This guarantees the existence of a k-th primitive root of unity in Z/pZ.

Procedures:

D.4.34.1 genSymId		compute ideal J such that sigma(J) = J and J includes I
D.4.34.2 isSymmetric		check if I is invariant under permutation sigma
D.4.34.3 primRoot		int describing a k-th primitive root of unity in Z/pZ
D.4.34.4 eigenvalues		list of eigenvalues of generators of I under sigma
D.4.34.5 symmStd		standard basis of I using invariance of I under sigma
D.4.34.6 syModStd		SB of I using modular methods and sigma(I) = I