  We give a self-contained proof of the strongest version of Mason's
conjecture, namely that for any matroid the sequence of the number of
independent sets of given sizes is ultra log-concave. To do this, we introduce
a class of polynomials, called completely log-concave polynomials, whose
bivariate restrictions have ultra log-concave coefficients. At the heart of our
proof we show that for any matroid, the homogenization of the generating
polynomial of its independent sets is completely log-concave.
