Mahler measures of polynomials that are sums of a bounded number of monomials

TL;DR

This paper investigates Mahler measures of polynomials with a limited number of monomials, establishing lower bounds, topological properties of their measure sets, and exploring coefficient determination for measure one cases.

Contribution

It provides new bounds on Mahler measures for sparse polynomials and analyzes the structure of their measure sets, especially for integer coefficient polynomials.

Findings

Mahler measure of such polynomials is at least h/2^{k-2}

The set of measures for integer coefficient polynomials is closed, with zero as an isolated point

Polynomials with Mahler measure 1 are partially determined by their coefficients

Abstract

We study Laurent polynomials in any number of variables that are sums of at most $k$ monomials. We first show that the Mahler measure of such a polynomial is at least $h/2^{k-2}$, where $h$ is the height of the polynomial. Next, restricting to such polynomials having integer coefficients, we show that the set of logarithmic Mahler measures of the elements of this restricted set is a closed subset of the nonnegative real line, with $0$ being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure $1$ is determined by its $k$ coefficients.