Clustering Complex Zeros of Triangular Systems of Polynomials

TL;DR

This paper introduces the first algorithm for clustering complex zeros of triangular polynomial systems within a specified region, leveraging recent univariate root clustering methods, and demonstrates its effectiveness through implementation and comparison with existing solvers.

Contribution

It presents a novel, certified subdivision-based algorithm for clustering zeros of multivariate polynomial systems, extending univariate root clustering techniques to higher dimensions.

Findings

Algorithm is correct and reliable.

Often faster than existing homotopy solvers.

Effective in practical experiments.

Abstract

This paper gives the first algorithm for finding a set of natural $\epsilon$-clusters of complex zeros of a triangular system of polynomials within a given polybox in $\mathbb{C}^n$, for any given $\epsilon>0$. Our algorithm is based on a recent near-optimal algorithm of Becker et al (2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is numeric, certified and based on subdivision. We implemented it and compared it with two well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solver that often gives correct answers, and sometimes faster than the one that gives sometimes correct results.