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

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, but  not  on the degree. In one variable the lower bound generalizes a
classical inequality of Mahler. In M variables the inequality depends on Z^M as an
ordered group, and in general the lower bound depends on the choice of ordering. In
one variable the proof is elementary. In M variables the proof exploits an idea used
in earlier work of D. Boyd. The talk should be accessible to a general mathematical
audience. This is joint work with S. Akhtari.