Upper Bounds on Polynomial Root Separation

TL;DR

This paper establishes an upper bound on the separation of polynomial roots in terms of the Mahler measure, providing a near-optimal estimate and exploring the impact of additional root constraints.

Contribution

It introduces the first known upper bound on root separation based on Mahler measure, improving understanding of polynomial root behavior.

Findings

Proves that root separation is bounded above by a function of degree and Mahler measure.

Shows the bound is sharp up to a constant factor.

Analyzes how additional assumptions like real roots affect the constant factor.

Abstract

In this paper, we consider the relationship between the Mahler measure of a polynomial and its separation. In 1964, Mahler proved that if $f(x) \in \mathbb{Z}[x]$ is separable of degree $n$, then $\operatorname{sep}(f) \gg_n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relation, and it led to questions about the optimal exponent on $M(f)$ in that relation. However, there has been relatively little study concerning upper bounds on $\operatorname{sep}(f)$ in terms of $M(f)$. In this paper, we prove that if $f(x) \in \mathbb{C}[x]$ has degree $n$, then $\operatorname{sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$, for example, if it only has real roots.