Upper Bounds on Polynomial Root Separation

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 $\mathrm{sep}(f){\gg }_{n}M(f{)}^{-(n-1)}$. 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 $\mathrm{sep}(f)$ in terms of $M(f)$. In this paper, we prove that if $f(x)\in \mathbb{C}[x]$ has degree $n$, then $\mathrm{sep}(f)\ll n^{-1/2}M(f{)}^{1/(n-1)}$. 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.

11 pages