### D.4.28 symodstd_lib

Library:
  symodstd.lib
Purpose:
  Procedures for computing Groebner basis of ideals being invariant under certain variable permutations.

Author:
  Stefan Steidel, steidel@mathematik.uni-kl.de

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 existance of a k-th primitive root of unity in Z/pZ.

Procedures:

> D.4.28.1 genSymId |  | compute ideal J such that sigma(J) = J and J includes I
> D.4.28.2 isSymmetric |  | check if I is invariant under permutation sigma
> D.4.28.3 primRoot |  | int describing a k-th primitive root of unity in Z/pZ
> D.4.28.4 eigenvalues |  | list of eigenvalues of generators of I under sigma
> D.4.28.5 symmStd |  | standard basis of I using invariance of I under sigma
> D.4.28.6 syModStd |  | SB of I using modular methods and sigma(I) = I

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