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A Note on Lawton’s Theorem

  • Edward Dobrowolski (a1)

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.

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[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]
[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
[3] Hajos, G., Solution of problem 41. Mat. Lapok 4(1953), 4041.
[4] Issa, Z. and Lalin, M., A generalization of a theorem ofBoyd and Lawton. Canad. Math. Bull. 56(2013), no. 4, 759768. http://dx.doi.Org/10.41 53/CMB-2O12-010-2
[5] Lawton, W. M., A problem ofBoyd concerning geometric means of polynomials. J. Number Theory 16(1983), no. 3, 356362. http://dx.doi.Org/1 0.101 6/0022-314X(83)90063-X
[6] Schmidt, K., Dynamical systems of algebraic origin. Progress in Mathematics, 128, Birkhauser Verlag, Basel, 1995.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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