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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.