Given a model in algebraic statistics and 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.

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.

We show that the complexity of the Markov bases of multidimensional tables stabilizes eventually if a single table dimension is allowed to vary. In particular, if this table dimension is greater than a computable bound, the Markov bases consist of elements from Markov bases of smaller tables. We give an explicit formula for this bound in terms of Graver bases. We also compute these Markov and Graver complexities for all K × 2 × 2 × 2 tables.

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gro¨bner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further, we show how to build Gro¨bner bases of a reducible model from the Gro¨bner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Gro¨bner bases. Finally, we present the solution of a problem posed by Vlach (Discrete Appl. Math. 13 (1986) 61-78) concerning the dimension of fibers coming from models corresponding to the boundary of a simplex. # 2002 Elsevier Science (USA)

The convex hull of the roots of a classical root lattice is called a root polytope. We determine explicit unimodular triangulations of the boundaries of the root polytopes associated to the root lattices An, Cn, and Dn, and we compute their f -and h-vectors. This leads us to recover formulae for the growth series of these root lattices, which were first conjectured by Conway, Mallows, and Sloane and Baake and Grimm and were proved by Conway and Sloane and Bacher, de la Harpe, and Venkov. We also prove the formula for the growth series of the root lattice Bn, which requires a modification of our technique. Introduction.A lattice L is a discrete subgroup of R n for some n ∈ Z >0 . The rank of a lattice is the dimension of the subspace spanned by the lattice. We say that a lattice L is generated as a monoid by a finite collection of vectors M = {a 1 , . . . , a r } if each u ∈ L is a nonnegative integer combination of the vectors in M. For convenience, we often write the vectors from M as columns of a matrix M ∈ R n×r , and to make the connection between L and M more transparent, we refer to the lattice generated by M as L M . The word length of u with respect to M, denoted w(u), is min( c i ) taken over all expressions u = c i a i with c i ∈ Z ≥0 . The growth function S(k) counts the number of elements u ∈ L with word length w(u) = k with respect to M. We define the growth series to be the generating function G(x) := k≥0 S(k) x k . It is a