A sharp Bombieri inequality, logarithmic energy and well conditioned polynomials

TL;DR

This paper investigates the relationships between polynomial condition numbers, spherical logarithmic energy minimizers, and establishes a sharp Bombieri inequality for complex polynomials, revealing new insights into polynomial conditioning and energy minimization.

Contribution

It demonstrates that optimally conditioned polynomials generate low-energy spherical point configurations and proves a new sharp Bombieri inequality for complex polynomials.

Findings

Polynomials with optimal condition number produce spherical points with small logarithmic energy.

A reverse relationship between energy minimizers and polynomial conditioning is established.

A new sharp Bombieri inequality for univariate complex polynomials is proved.

Abstract

In this paper we explore the connections between minimizers of the discrete logarithmic energy on the 2-dimensional sphere, univariate polynomials with optimal condition number in the Shub-Smale sense and a quotient involving norms of polynomials. Our main results are that polynomials with optimal condition number produce spherical points with small logarithmic energy (the reverse result was already known) and a sharp Bombieri type inequality for univariate polynomials with complex coefficients.