Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling

TL;DR

This paper introduces an efficient algorithm for counting polynomial roots within convex regions using winding number calculations, suitable for high-precision polynomial evaluation scenarios.

Contribution

It presents a novel algorithm for winding number computation that improves root counting in convex regions, with proofs and complexity analysis.

Findings

Algorithm effectively counts roots in convex regions

Works with lower-precision polynomial evaluations

Includes correctness proof and complexity estimation

Abstract

In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and exclusion tests in a subdivision algorithms for polynomial root-finding, and would be especially usefull in application scenarios where high-precision polynomial coefficients are hard to obtain but we succeed with counting already by using polynomial evaluation with lower precision. We provide the pseudo code of the algorithm, proof of its correctness as well as estimation of its complexity.