Stable normal forms for polynomial system solving

TL;DR

This paper introduces a generalized border basis algorithm for polynomial system solving that relaxes monomial ordering constraints, provides stable normal forms under perturbations, and enhances practical efficiency demonstrated through benchmarks.

Contribution

It presents a novel border basis algorithm that broadens the formalism for quotient algebra representations and proves stability of normal forms under small ideal perturbations.

Findings

Algorithm extends existing border basis methods.

Normal forms are stable under small perturbations.

Experimental results show improved efficiency on benchmarks.

Abstract

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending the one provided in [MT'05]. This general border basis algorithm weakens the monomial ordering requirement for \grob bases computations. It is up to date the most general setting for representing quotient algebras, embedding into a single formalism Gr\"obner bases, Macaulay bases and new representation that do not fit into the previous categories. With this formalism we show how the syzygies of the border basis are generated by commutation relations. We also show that our construction of normal form is stable under small perturbations of the ideal, if the number of solutions remains constant. This new feature for a symbolic algorithm has a huge impact on the practical efficiency as it is illustrated by the experiments on classical benchmark polynomial systems, at the end of the paper.