  Maximum likelihood estimation in statistics leads to the problem of
maximizing a product of powers of polynomials. We study the algebraic degree of
the critical equations of this optimization problem. This degree is related to
the number of bounded regions in the corresponding arrangement of
hypersurfaces, and to the Euler characteristic of the complexified complement.
Under suitable hypotheses, the maximum likelihood degree equals the top Chern
class of a sheaf of logarithmic differential forms. Exact formulae in terms of
degrees and Newton polytopes are given for polynomials with generic
coefficients.
