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UPPER BOUNDS ON POLYNOMIAL ROOT SEPARATION

Part of: Real and complex fields Algebraic number theory: global fields Geometric function theory

Published online by Cambridge University Press:  20 January 2025

GREG KNAPP Open the ORCID record for GREG KNAPP [Opens in a new window]  and
CHI HOI YIP Open the ORCID record for CHI HOI YIP [Opens in a new window]
GREG KNAPP*
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta T2N 1N4, Canada
CHI HOI YIP
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA e-mail: [email protected]
*
e-mail: [email protected]
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Abstract

We consider the relationship between the Mahler measure $M(f)$ of a polynomial f and its separation $\operatorname {sep}(f)$. Mahler [‘An inequality for the discriminant of a polynomial’, Michigan Math. J. 11 (1964), 257–262] 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 relationship and led to questions about the optimal exponent on $M(f)$. However, there has been relatively little study concerning upper bounds on $\operatorname {sep}(f)$ in terms of $M(f)$. 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 has only real roots.

Keywords

polynomialMahler measureroot separation

MSC classification

Primary: 12D10: Polynomials
Secondary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , First View , pp. 1 - 16
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first author gratefully acknowledges the financial support of a PIMS postdoctoral fellowship, along with NSERC grants RGPIN-2019-04844, RGPIN-2022-03559 and RGPIN-2018-03770. The research of the second author was supported in part by an NSERC fellowship.

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