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Proceedings of the American Mathematical Society

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On Mahler's measure of a polynomial


Author: Alain Durand
Journal: Proc. Amer. Math. Soc. 83 (1981), 75-76
MSC: Primary 30C10
DOI: https://doi.org/10.1090/S0002-9939-1981-0619985-1
MathSciNet review: 619985
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Abstract: Let $ P$ be a polynomial with complex coefficients. We denote by $ M(P)$ the Mahler measure of $ P$ (resp. the maximum modulus of $ P$ on the disk $ \left\vert z \right\vert \leqslant 1)$). We prove here that $ M(P) = \inf \left\Vert {PQ} \right\Vert$ where the infimum is taken over all polynomials $ Q$ with complex coefficients satisfying $ Q(0) = 1$.


References [Enhancements On Off] (What's this?)

  • [1] W. Lawton, Heights of algebraic numbers and Szegö's theorem, Proc. Amer. Math. Soc. 49 (1975), 47-50. MR 0376628 (51:12803)
  • [2] K. Mahler, An application of Jensen's formula to polynomials, Mathematika 7 (1960), 98-100. MR 0124467 (23:A1779)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0619985-1
Keywords: Mahler's measure, Jensen's theorem, maximum modulus principle
Article copyright: © Copyright 1981 American Mathematical Society

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