A signature-based algorithm for computing the nondegenerate locus of a polynomial system

• Published in: Journal of symbolic computation Vol. 119; pp. 1 - 21
• Main Authors: Eder, Christian, Lairez, Pierre, Mohr, Rafael, El Din, Mohab Safey
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.11.2023; Elsevier
• Subjects: Algorithm; Computer Science; Gröbner basis; Ideal decomposition; Symbolic Computation; Gröbner basis; Algorithm; Ideal decomposition; Gröbner Basis; Ideal Decomposition
• ISSN: 0747-7171; 1095-855X
• DOI: 10.1016/j.jsc.2023.02.001

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.