Upper Bounds on Polynomial Root Separation

2024/10/01 by Knapp, Greg, Yip, Chi Hoi
#11H99 #11R06 (Primary) #12D10 (Secondary) #30C15 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.01126

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) ∈ ℤ[x] is separable of degree n, then sep(f) ≫_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 sep(f) in terms of M(f). In this paper, we prove that if f(x) ∈ ℂ[x] has degree n, then sep(f) ≪ n^-1/2M(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.

Related

• Generalizing The Davenport-Mahler-Mignotte Bound -- The Weighted Case
• Polynomial minimum root separation
• The alternative to Mahler measure of a multivariate polynomial
• Reprint: An inequality for the discriminant of a polynomial (1964)
• Root separation for irreducible integer polynomials