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arXiv · 2404.13514

A new iterative algorithm for comprehensive Grobner systems

Anna Maria Bigatti·Elisa Palezzato·Michele Torielli
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Abstract

A Comprehensive Grobner system for a parametric ideal I in K(A)[X] represents the collection of all Grobner bases of the ideals I' in K[X] obtained as the values of the parameters A vary in K. The recent algorithms for computing them comprehensive Grobner systems consider the corresponding ideal J in K[A,X], and are based on stability of Grobner bases of ideals under specializations of the parameters. Starting from a Grobner basis of J, the computation splits recursively depending on the vanishing of the evaluation of some ``coefficients'' in K[A]. In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Grobner systems. We show how we keep track of the sub-cases to be considered, and how we avoid some redundant computation branches using ``comparatively-cheap'' ideal-membership tests, instead of radical-membership tests.

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Anna Maria Bigatti, Elisa Palezzato, Michele Torielli. 2024-04-21. A new iterative algorithm for comprehensive Grobner systems. https://arxiv.org/abs/2404.13514

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