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arXiv · 2202.13784

A Signature-based Algorithm for Computing the Nondegenerate Locus of a Polynomial System

Christian Eder·Pierre Lairez·Rafael Mohr·Mohab Safey El Din
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Abstract

Polynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The nondegenerate locus of a polynomial system is the set of points where the codimension of the solution set matches the number of equations. Computing the nondegenerate locus is classically done through ideal-theoretic operations in commutative algebra such as saturation ideals or equidimensional decompositions to extract the component of maximal codimension. By exploiting the algebraic features of signature-based Gr\"obner basis algorithms we design an algorithm which computes a Gr\"obner basis of the equations describing the closure of the nondegenerate locus of a polynomial system, without computing first a Gr\"obner basis for the whole polynomial system.

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Christian Eder, Pierre Lairez, Rafael Mohr, Mohab Safey El Din. 2022-02-28. A Signature-based Algorithm for Computing the Nondegenerate Locus of a Polynomial System. https://doi.org/10.1016/j.jsc.2023.02.001

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