Deterministically Computing Reduction Numbers of Polynomial Ideals

Co-author(s): Amir Hashemi and Michael Schweinfurter

Reference: Computer Algebra in Scientific Computing - CASC 2014, V.P. Gerdt, W. Koepf, W.M. Seiler, E.V. Vorozhtsov (eds.), Springer-Verlag, Berlin 2014, Lecture Notes in Computer Science 8660, pp. 186-201

Description: We study how to compute the absolute and the big reduction number of a polynomial ideal and the set of all possible reduction numbers. The emphasis is on deterministic approaches that do not apply probabilistic methods (for example to compute the generic initial ideal). For the absolute reduction number, we add sufficiently generic linear forms and then compute the regularity of the obtained reduction with Pommaret bases. For computing the reduction number set (and thus also the big reduction number), we use Gröbner systems. While computationally rather expensive, we believe that this approach is still much more efficient than alternative approaches proposed in the literature.

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