Serkan Hosten

The maximum likelihood degree

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.

Solving the Likelihood Equations

AbstractGiven 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.

Gröbner bases and polyhedral geometry of reducible and cyclic models

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Grobner 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 Grobner bases of a reducible model from the Grobner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Grobner 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.

A finiteness theorem for Markov bases of hierarchical models

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 Kx2x2x2 tables.

Computing the integer programming gap

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.

Standard pairs and group relaxations in integer programming

Abstract The main result of this paper is a non-Buchberger algorithm for constructing initial ideals and Grobner bases of toric ideals, based on the connections between toric ideals and integer programming. The tools used are those of standard pair decompositions of standard monomials of a toric initial ideal, localizations of such ideals at their associated primes and group relaxations of integer programs. We give an algorithm for constructing standard pair decompositions, provide degree bounds for certain elements in the reduced Grobner bases of toric ideals, and derive bounds on the arithmetic degree of initial ideals of monomial curves. We also exhibit new results for the localizations of initial ideals arising from toric ideals of codimension two.

The order dimension of the complete graph

Abstract We show that the order dimension of the complete graph on n vertices is the smallest integer t for which there are n antichains in the subset lattice of [ t - 1] that do not contain [t - 1] or two sets whose union is [ t - 1].

Root Polytopes and Growth Series of Root Lattices

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

Gomory integer programs

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Computational algebra for bifurcation theory

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