  Given a model in algebraic statistics and some data, the likelihood function
is a rational function on a projective variety. Algebraic algorithms are
presented for computing all critical points of this function, with the aim of
identifying the local maxima in the probability simplex. Applications include
models specified by rank conditions on matrices and the Jukes-Cantor models of
phylogenetics. The maximum likelihood degree of a generic complete intersection
is also determined.
