2003/11/15 by Christopher D. Sinclair, Sinclair, Christopher D.
Materials Science · Mathematics · #33E20 #44A05 #FOS: Mathematics #Fractional Differential Equations Solutions #Liquid Crystal Research Advancements #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT #msc:33E20 #msc:44A05
paper · pdf · doi:10.48550/arxiv.math/0311255
13 pages. To be published in Int. J. Math. Math. Sci
arxiv created 2003/11/15 · openalex publication_date 2003/11/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 anti-reciprocal) 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.