A condition number analysis of an algorithm for solving a system of polynomial equations with one degree of freedom

TL;DR

This paper introduces a condition number framework for analyzing an algorithm based on Newton's method and subdivision, designed to solve nondegenerate systems of polynomial equations with one degree of freedom, with runtime bounds depending on this condition number.

Contribution

It defines a new condition number for polynomial systems that measures proximity to degeneracy and analyzes the algorithm's runtime in terms of this condition number.

Findings

Condition number distinguishes well-conditioned from ill-conditioned instances.

Algorithm's runtime is bounded by the condition number, polynomial degrees, and number of variables.

Framework excludes degenerate cases with infinite ill-conditioning.

Abstract

This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate cases, in which case the solution is a 1-dimensional curve. Our first main contribution is a definition of a condition number measuring reciprocal distance to degeneracy that can distinguish poor and well conditioned instances of this problem. (Degenerate problems would be infinitely ill conditioned in our framework.) Our second contribution, which is the main novelty of our algorithm, is an analysis showing that its running time is bounded in terms of the condition number of the problem instance as well as $n$ and the polynomial degrees.