Multiplicative distance functions

We generalize Mahler’s measure to create the class of multiplicative distance functions on C[x]. These functions are uniquely determined by their action on the roots of polynomials. We find a simple asymptotic condition that determines which functions on C are induced by multiplicative distance functions, and use this to give several examples. In particular, we show how Mahler’s measure restricted to the set of reciprocal polynomials may be viewed as a multiplicative distance function: the reciprocal Mahler’s measure. We then turn to potential theory to demonstrate how new multiplicative distance functions may be created by generalizing Jensen’s formula. In so doing we will introduce multiplicative distance functions which measure the complexity of polynomials in C[x] by comparing the geometry of their roots to compact subsets of C. Let s be a complex variable, and let N be a positive integer. To every multiplicative distance function Φ we will define an analytic function FN (Φ; s) (HN (Φ; s) resp.) which encodes information about the range of values Φ takes on degree N polynomials in R[x] (C[x] resp.). These functions are analytic in the half plane (s) > N. We show that HN (s) can be represented as the determinant of a Gram matrix in a Hilbert space dependent on s and Φ. This revelation allows us to write HN (s) as the product of the norms of N vectors in the associated Hilbert space. Several examples are presented. Similarly, when N is even we introduce a skew-symmetric inner product associated to Φ and s and show that FN (s) can be written as the Pfaffian of an antisymmetric Gram matrix defined from this skew-symmetric inner product. This allows us to write FN (s) as a product of N/2 simpler functions of s. We use this information to compute FN (s) for the reciprocal Mahler’s measure, and in so doing discover that this function is an even rational function of s with rational coefficients and simple poles at small integers.