3 research outputs found
Groebner bases of symmetric ideals
In this article we present two new algorithms to compute the Groebner basis
of an ideal that is invariant under certain permutations of the ring variables
and which are both implemented in SINGULAR (cf. [DGPS12]). The first and major
algorithm is most performant over finite fields whereas the second algorithm is
a probabilistic modification of the modular computation of Groebner bases based
on the articles by Arnold (cf. [A03]), Idrees, Pfister, Steidel (cf. [IPS11])
and Noro, Yokoyama (cf. [NY12], [Y12]). In fact, the first algorithm that
mainly uses the given symmetry, improves the necessary modular calculations in
positive characteristic in the second algorithm. Particularly, we could, for
the first time even though probabilistic, compute the Groebner basis of the
famous ideal of cyclic 9-roots (cf. [BF91]) over the rationals with SINGULAR.Comment: 17 page