The distribution of Mahler's measures of reciprocal polynomials

Autor: Christopher D. Sinclair
Jazyk: angličtina
Rok vydání: 2004
Předmět:
Zdroj: International Journal of Mathematics and Mathematical Sciences, Vol 2004, Iss 52, Pp 2773-2786 (2004)
Druh dokumentu: article
ISSN: 0161-1712
1687-0425
01611712
DOI: 10.1155/S0161171204312469
Popis: 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.
Databáze: Directory of Open Access Journals
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