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

We study Laurent polynomials in any number of variables that are sums of at most [Formula: see text] monomials. We first show that the Mahler measure of such a polynomial is at least [Formula: see text], where [Formula: see text] 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 [Formula: see text] being an isolated point of the set. In the final section, we discuss the extent to which such an integer polynomial of Mahler measure [Formula: see text] is determined by its [Formula: see text] coefficients.