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 consider the corresponding ideal J in K[A, X], and are based on stability of Grobner bases of ideals under specializations of the parameters A. 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.
A new iterative algorithm for comprehensive Gröbner systems
Bigatti A. M.;
2025-01-01
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 consider the corresponding ideal J in K[A, X], and are based on stability of Grobner bases of ideals under specializations of the parameters A. 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.
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
