Lower bounds for Mahler measure that depend on the number of monomials

Abstract: We prove a new lower bound for the Mahler measure of a polynomial in one and in several variables that depends on the complex coefficients, and the number of monomials. In one variable our result generalizes a classical inequality of Mahler. In $M$ variables our result depends on $\mathbb{Z}^M$ as an ordered group, and in general our lower bound depends on the choice of ordering.