Gröbner bases of symmetric ideals. (English) Zbl 1277.13001

Summary: 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 [E. A. Arnold, J. Symb. Comput. 35, No. 4, 403-419 (2003; Zbl 1046.13018); N. Idrees, G. Pfister and S. Steidel, J. Symb. Comput. 46, No. 6, 672–684 (2011; Zbl 1229.13002)] and M. Noro and K. Yokoyama [“Usage of modular techniques for efficient computation of ideal operation”, in preparation (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. [G. Björck and R. Fröberg, J. Symb. Comput. 12, 329–336 (1991; Zbl 0751.12001)]) over the rationals with Singular.


13-04 Software, source code, etc. for problems pertaining to commutative algebra
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
68W30 Symbolic computation and algebraic computation
13F20 Polynomial rings and ideals; rings of integer-valued polynomials


Full Text: DOI arXiv


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