Deterministically computing reduction numbers of polynomial ideals. (English) Zbl 1416.68220
Gerdt, Vladimir P. (ed.) et al., Computer algebra in scientific computing. 16th international workshop, CASC 2014, Warsaw, Poland, September 8–12, 2014. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 8660, 186-201 (2014).
Summary: We discuss the problem of determining reduction numbers of a polynomial ideal \(\mathcal{I}\) in \(n\) variables. We present two algorithms based on parametric computations. The first one determines the absolute reduction number of \(\mathcal{I}\) and requires computations in a polynomial ring with \((n-\dim \mathcal{I})\dim \mathcal{I}\) parameters and \(n - \dim \mathcal{I}\) variables. The second one computes via a Gröbner system the set of all reduction numbers of the ideal \(\mathcal{I}\) and thus in particular also its big reduction number. However, it requires computations in a ring with \(n \dim \mathcal{I}\) parameters and \(n\) variables.
For the entire collection see [Zbl 1295.68022].
For the entire collection see [Zbl 1295.68022].
MSC:
68W30 | Symbolic computation and algebraic computation |
13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |