A Stabilized Normal Form Algorithm for Generic Systems of Polynomial Equations

TL;DR

This paper introduces a numerically stable algorithm for computing normal forms of zero-dimensional polynomial systems, enabling efficient solutions by selecting optimal bases through linear algebra techniques.

Contribution

It presents an automated basis selection method for quotient rings in polynomial systems, improving the stability and feasibility of normal form computations in finite precision.

Findings

Automated basis selection enhances numerical stability.

Method effectively computes solutions for generic polynomial systems.

Applicable to systems with solutions equal to the Bézout number.

Abstract

We propose a numerical linear algebra based method to find the multiplication operators of the quotient ring $\mathbb{C}[x]/I$ associated to a zero-dimensional ideal $I$ generated by $n$ $\mathbb{C}$-polynomials in $n$ variables. We assume that the polynomials are generic in the sense that the number of solutions in $\mathbb{C}^n$ equals the B\'ezout number. The main contribution of this paper is an automated choice of basis for $\mathbb{C}[x]/I$, which is crucial for the feasibility of normal form methods in finite precision arithmetic. This choice is based on numerical linear algebra techniques and governed by the numerical properties of the given generators of $I$.