The range of multiplicative functions on C[x], R[x] and Z[x]

Abstract

Mahler's measure is generalized to create the class of {\it multiplicative distance functions}. These functions measure the complexity of polynomials based on the location of their zeros in the complex plane. Following work of S.-J. Chern and J. Vaaler in \cite{chern-vaaler}, we associate to each multiplicative distance function two families of analytic functions which encode information about its range on \C[x] and \R[x]. These {\it moment functions} are Mellin transforms of distribution functions associated to the multiplicative distance function and demonstrate a great deal of arithmetic structure. For instance, we show that the moment function associated to Mahler's measure restricted to real reciprocal polynomials of degree 2N has an analytic continuation to rational functions with rational coefficients, simple poles at integers between -N and N, and a zero of multiplicity 2N at the origin. This discovery leads to asymptotic estimates for the number of reciprocal integer polynomials of fixed degree with Mahler measure less than T as $T \to \infty$. To explain the structure of this moment functions we show that the real moment functions of a multiplicative distance function can be written as Pfaffians of antisymmetric matrices formed from a skew-symmetric bilinear form associated to the multiplicative distance function.