Hostname: page-component-6bf8c574d5-zc66z Total loading time: 0 Render date: 2025-03-09T02:24:39.389Z Has data issue: false hasContentIssue false
  • English
  • Français

UPPER BOUNDS ON POLYNOMIAL ROOT SEPARATION

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

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: cyip30@gatech.edu
*
e-mail: greg.knapp@ucalgary.ca
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.

References

Albayrak, S., Ghosh, S., Knapp, G. and Nguyen, K. D., ‘On certain polytopes associated to products of algebraic integer conjugates’, Preprint, 2024, arXiv:2408.00250.CrossRefGoogle Scholar
Berndt, B. C., Kim, S. and Zaharescu, A., ‘The circle problem of Gauss and the divisor problem of Dirichlet—Still unsolved’, Amer. Math. Monthly 125(2) (2018), 99–114.CrossRefGoogle Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine Geometry, New Mathematical Monographs, 4 (Cambridge University Press, Cambridge, 2006).Google Scholar
Bugeaud, Y. and Dujella, A., ‘Root separation for irreducible integer polynomials’, Bull. Lond. Math. Soc. 43(6) (2011), 1239–1244.CrossRefGoogle Scholar
Bugeaud, Y. and Dujella, A., ‘Root separation for reducible integer polynomials’, Acta Arith. 162(4) (2014), 393–403.CrossRefGoogle Scholar
Bugeaud, Y., Dujella, A., Fang, W., Pejković, T. and Salvy, B., ‘Absolute root separation’, Exp. Math. 31(3) (2022), 806–813.CrossRefGoogle Scholar
Bugeaud, Y., Dujella, A., Pejković, T. and Salvy, B., ‘Absolute real root separation’, Amer. Math. Monthly 124(10) (2017), 930–936.CrossRefGoogle Scholar
Dujella, A. and Pejković, T., ‘Root separation for reducible monic polynomials of odd degree’, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 21(532) (2017), 21–27.Google Scholar
Evertse, J.-H., ‘Distances between the conjugates of an algebraic number’, Publ. Math. Debrecen 65(3–4) (2004), 323–340.CrossRefGoogle Scholar
Grundman, H. G. and Wisniewski, D. P., ‘Tetranomial Thue equations’, J. Number Theory 133(12) (2013), 4140–4174.CrossRefGoogle Scholar
Johnson, R., ‘Upper limit on the central binomial coefficient’, MathOverflow. https://mathoverflow.net/q/384815 (version: 24 February 2021).Google Scholar
Knapp, G., Polynomial Root Distribution and its Impact on Solutions to Thue Equations. PhD Thesis (University of Oregon, 2023).Google Scholar
Koiran, P., ‘Root separation for trinomials’, J. Symbolic Comput. 95 (2019), 151–161.CrossRefGoogle Scholar
Lehmer, D. H., ‘Factorization of certain cyclotomic functions’, Ann. of Math. (2) 34(3) (1933), 461–479.CrossRefGoogle Scholar
Mahler, K., ‘An inequality for the discriminant of a polynomial’, Michigan Math. J. 11 (1964), 257–262.CrossRefGoogle Scholar
Robbins, H., ‘A remark on Stirling’s formula’, Amer. Math. Monthly 62 (1955), 26–29.Google Scholar
Rump, S. M., ‘Polynomial minimum root separation’, Math. Comp. 33(145) (1979), 327–336.CrossRefGoogle Scholar
Smyth, C., ‘The Mahler measure of algebraic numbers: a survey’, in: Number Theory and Polynomials, London Mathematical Society Lecture Note Series, 352 (Cambridge University Press, Cambridge, 2008), 322–349.CrossRefGoogle Scholar
Wendel, J. G., ‘Note on the gamma function’, Amer. Math. Monthly 55 (1984), 563–564.CrossRefGoogle Scholar