Certified counting of roots of random univariate polynomials

TL;DR

This paper presents a combined approach using multiprecision computation and certification methods to reliably count real roots of high-degree random univariate polynomials, with new experimental insights.

Contribution

It introduces a novel combination of certification techniques with efficient computation to accurately count roots of random polynomials, including new results for Cauchy-distributed coefficients.

Findings

Successful certification of root counts for high-degree polynomials

Comparison of Smale's α-theory and Gerschgorin-based certification methods

New experimental results for polynomials with Cauchy-distributed coefficients

Abstract

A challenging problem in computational mathematics is to compute roots of a high-degree univariate random polynomial. We combine an efficient multiprecision implementation for solving high-degree random polynomials with two certification methods, namely Smale's $\alpha$-theory and one based on Gerschgorin's theorem, for showing that a given numerical approximation is in the quadratic convergence region of Newton's method of some exact solution. With this combination, we can certifiably count the number of real roots of random polynomials. We quantify the difference between the two certification procedures and list the salient features of both of them. After benchmarking on random polynomials where the coefficients are drawn from the Gaussian distribution, we obtain novel experimental results for the Cauchy distribution case.