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Two sharp inequalities for the norm of a factor of a polynomial

Part of: Geometric function theory

Published online by Cambridge University Press:  26 February 2010

David W. Boyd
David W. Boyd
Affiliation:
Department of Mathematics, The University of British Columbia, 121-1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Y4.
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Abstract

Let f(x) be a monic polynomial of degree n with complex coefficients, which factors as f(x) = g(x)h(x), where g and h are monic. Let be the maximum of on the unit circle. We prove that , where β = M(P0) = 1 38135 …, where P0 is the polynomial P0(x, y) = 1 + x + y and δ = M(P1) = 1 79162…, where P1(x, y) = 1 + x + y - xy, and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.

MSC classification

Secondary: 30C10: Polynomials
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 341 - 349
Copyright
Copyright © University College London 1992

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References

[AS]Abramowitz, M. and Stegun, I. A.. “Handbook of Mathematical Functions” (Dover, N.Y., 1965).Google Scholar
[BE]Beauzamy, B. and Enflo, P.. Estimations de produits de polynômes. J Number Theory, 21 (1985), 390–412.CrossRefGoogle Scholar
[BBEM]Beauzamy, B.Bombieri, E.Enflo, P. and Montgomery, H. L.. Products of Polynomials in Many Variables. J. Number Theory, 36 (1990), 219–245.CrossRefGoogle Scholar
[B1]Boyd, D. W.. Speculations Concerning the Range of Mahler's Measure. Canad. Math. Bull., 24 (1981), 453–469.CrossRefGoogle Scholar
[B2]Boyd, D. W.. The Asymptotic Behaviour of the Binomial Circulant Determinant. J. Math. Anal. Appl., 86 (1982), 30–38.CrossRefGoogle Scholar
[EM]Evans, R. and Montgomery, P.. Some Unimodal Reciprocal Polynomials with Positive Coefficients, Advanced Problem 6631. Amer. Math. Monthly, 98 (1991), 870–872.CrossRefGoogle Scholar
[Fr]Frame, J. S.. Factors of the binomial circulant determinant. Fibronacci Quart., 18 (1980), 9–23.Google Scholar
[Ge]Gelfond, A. O.. Transcendental and Algebraic Numbers (Dover, N.Y., 1960). Translation by Boron, L. F. of the 1952 Russian edition.Google Scholar
[G1]Glesser, P.. Nouvelle Majoration de la norme des Facteurs d'un Polynôme. C. R. Math. Rep. Acad. Sci. Canada, 12 (1990), 224–228.Google Scholar
[Gr]Granville, A.. Bounding the Coefficients of a Divisor of a Given Polynomial. Monatsh. Math., 109 (1990), 271–277.CrossRefGoogle Scholar
[Le]Lewin, L.. Polylogarithms and Associated Functions (Elsevier North-Holland, Amsterdam, 1981).Google Scholar
[LSW]Lind, D.Schmidt, K. and Ward, T.. Mahler measure and entropy for commuting automorphisms of compact groups. Invent. Math., 101 (1990), 593–629.CrossRefGoogle Scholar
[M1]Mahler, K.. An application of Jensen's Formula to Polynomials. Mathematika, 7 (1960), 98–100.CrossRefGoogle Scholar
[M2]Mahler, K.. On Some Inequalities for Polynomials in Several Variables. J. Land. Math. Soc., 37 (1962), 341–344.CrossRefGoogle Scholar
[M3]Mahler, K.. A Remark on a Paper of Mine on Polynomials. Illinois J. Math., 8 (1964), 1–4.CrossRefGoogle Scholar
[Mig]Mignotte, M.. Some useful bounds. In: Buchberger, B. et al. (eds), Computer Algebra, Symbolic and Algebraic Computation (Springer, N.Y., 1982), 259–263.Google Scholar
[Mil]Milnor, J.. Hyperbolic Geometry: The First 150 Years. Bull. Amer. Math. Soc. (N.S.), 6 (1982), 9–24.CrossRefGoogle Scholar