Bounds for the maximum modulus of polynomial roots with nearly optimal worst-case overestimation

TL;DR

This paper establishes a nearly optimal upper bound for the maximum modulus of polynomial roots, addressing limitations of previous bounds that were often tested on limited examples or classes.

Contribution

It provides a simple, nearly optimal worst-case overestimation bound for polynomial root moduli, improving upon existing bounds.

Findings

The bound is nearly optimal in worst-case scenarios.

Previous bounds were limited to specific examples or classes.

The new bound offers a more reliable estimate across general polynomials.

Abstract

Many upper bounds for the moduli of polynomial roots have been proposed but reportedly assessed on selected examples or restricted classes only. Regarding quality measured in terms of worst-case relative overestimation of the maximum root-modulus we establish a simple, nearly optimal result.