## Multiplicative distance functions

##### Abstract

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.

##### Department

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