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International Journal of Mathematics and Mathematical Sciences
Volume 2004, Issue 52, Pages 2773-2786
http://dx.doi.org/10.1155/S0161171204312469

The distribution of Mahler's measures of reciprocal polynomials

Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin 78712 025, TX, USA

Received 29 May 2003; Revised 4 December 2003

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the distribution of Mahler's measures of reciprocal polynomials with complex coefficients and bounded even degree. We discover that the distribution function associated to Mahler's measure restricted to monic reciprocal polynomials is a reciprocal (or antireciprocal) Laurent polynomial on [1,) and identically zero on [0,1). Moreover, the coefficients of this Laurent polynomial are rational numbers times a power of π. We are led to this discovery by the computation of the Mellin transform of the distribution function. This Mellin transform is an even (or odd) rational function with poles at small integers and residues that are rational numbers times a power of π. We also use this Mellin transform to show that the volume of the set of reciprocal polynomials with complex coefficients, bounded degree, and Mahler's measure less than or equal to one is a rational number times a power of π.