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

TL;DR

This paper introduces a signature-based algorithm that efficiently computes the nondegenerate locus of a polynomial system by avoiding full Gr"obner basis computations, thus improving algebraic geometric analysis.

Contribution

The paper presents a novel signature-based approach to compute the nondegenerate locus without full ideal decomposition, enhancing computational efficiency.

Findings

Algorithm successfully computes the nondegenerate locus

Reduces computational complexity compared to classical methods

Applicable to various polynomial systems in geometric modeling

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.