Journal of Symbolic Computation
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Modular algorithms for computing Gröbner bases
Journal of Symbolic Computation
Algebraic Statistics for Computational Biology
Algebraic Statistics for Computational Biology
A Singular Introduction to Commutative Algebra
A Singular Introduction to Commutative Algebra
Solving systems of polynomial equations with symmetries using SAGBI-Gröbner bases
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Parallelization of Modular Algorithms
Journal of Symbolic Computation
Usage of modular techniques for efficient computation of ideal operations
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Gröbner bases of ideals invariant under a commutative group: the non-modular case
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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In this article we present two new algorithms to compute the Grobner 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 Grobner 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 Grobner basis of the famous ideal of cyclic 9-roots (cf. Bjorck and Froberg, 1991) over the rationals with Singular.