A Signature-based Algorithm for Computing the Nondegenerate Locus of a Polynomial System
arXiv:2202.13784 · doi:10.1016/j.jsc.2023.02.001
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 Gröbner basis algorithms we design an algorithm which computes a Gröbner basis of the equations describing the closure of the nondegenerate locus of a polynomial system, without computing first a Gröbner basis for the whole polynomial system.
22 pages, 2 figures. Substantial rewrite of content of the parts of the paper involving signature-based Gröbner basis algorithms, both the exposition and the description of the core algorithm of the paper changed