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<p>A comprehensive Gröbner system for a parametric ideal <i>I</i> in <i>K</i>(<i>A</i>)[<i>X</i>] represents the collection of all Gröbner bases of the ideals <i>I</i>′ in <i>K</i>[<i>X</i>] obtained as the values of the parameters <i>A</i> vary in <i>K</i>. The recent algorithms for computing them consider the corresponding ideal <i>J</i> in <i>K</i>[<i>A</i>,&#xa0;<i>X</i>], and are based on stability of Gröbner bases of ideals under specializations of the parameters <i>A</i>. Starting from a Gröbner basis of <i>J</i>, the computation splits recursively depending on the vanishing of the evaluation of some “coefficients” in <i>K</i>[<i>A</i>]. In this paper, taking inspiration from the algorithm described by Nabeshima, we create a new iterative algorithm to compute comprehensive Gröbner 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.</p>

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A new iterative algorithm for comprehensive Gröbner systems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月31日

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  • Anna Maria Bigatti,
  • Elisa Palezzato,
  • Michele Torielli

摘要

A comprehensive Gröbner system for a parametric ideal I in K(A)[X] represents the collection of all Gröbner 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 Gröbner bases of ideals under specializations of the parameters A. Starting from a Gröbner 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 Gröbner 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.