Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T20:48:59.307Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Lawton's Theorem

Published online by Cambridge University Press:  20 November 2018

Edward Dobrowolski
Edward Dobrowolski*
Affiliation:
Mathematics Department, University of Northern British Columbia, Prince George, BC. e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove Lawton's conjecture about the upper bound on themeasure of the set on the unit circle on which a complex polynomial with a bounded number of coefficients takes small values. Namely, we prove that Lawton's bound holds for polynomials that are not necessarily monic. We also provide an analogous bound for polynomials in several variables. Finally, we investigate the dependence of the bound on the multiplicity of zeros for polynomials in one variable.

Keywords

11R0911R06polynomialMahler measure
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 3 , 01 September 2017 , pp. 484 - 489
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Dobrowolski, E. and Smyth, C., Mahler measures of polynomials that are sums of a bounded number of monomials. arxiv:1 606.04376 [math.NT]Google Scholar
[2] Everest, G. and Ward, T., Heights of polynomials and entropy in algebraic dynamics. Universitext, Springer-Verlag, London, 1999. http://dx.doi.Org/10.1007/978-1-4471-3898-3 Google Scholar
[3] Hajos, G., Solution of problem 41. Mat. Lapok 4(1953), 40–41. Google Scholar
[4] Issa, Z. and Lalin, M., A generalization of a theorem ofBoyd and Lawton. Canad. Math. Bull. 56(2013), no. 4, 759–768. http://dx.doi.Org/10.41 53/CMB-2O12-010-2 Google Scholar
[5] Lawton, W. M., A problem ofBoyd concerning geometric means of polynomials. J. Number Theory 16(1983), no. 3, 356–362. http://dx.doi.Org/1 0.101 6/0022-314X(83)90063-X Google Scholar
[6] Schmidt, K., Dynamical systems of algebraic origin. Progress in Mathematics, 128, Birkhauser Verlag, Basel, 1995.Google Scholar