Gröbner bases of symmetric ideals

In this article we present two new algorithms to compute the Gröbner basis of an ideal that is invariant under certain permutations of the ring variables and which are both implemented in Singular (cf. Decker et al., 2012, ). The first and major algorithm is most performant over finite fields whereas the second algorithm is a probabilistic modification of the modular computation of Gröbner bases based on the articles by Arnold (cf. Arnold, 2003, ), Idrees, Pfister, Steidel (cf. Idrees et al., 2011, ) and Noro, Yokoyama (cf. Noro and Yokoyama, in preparation, ; Yokoyama, 2012, ). 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 Gröbner basis of the famous ideal of cyclic 9-roots (cf. Björck and Fröberg, 1991) over the rationals with Singular.