The Circuit Ideal of a Vector Configuration

The circuit ideal, \ica, of a configuration \A = \{\a_1, ..., \a_n\} \subset \Z^d is the ideal generated by the binomials {\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n] as \cc = \cc^+ - \cc^- \in \Z^n varies over the circuits of \A. This ideal is contained in the toric ideal, \ia, of \A which has numerous applications and is nontrivial to compute. Since circuits can be computed using linear algebra and the two ideals often coincide, it is worthwhile to understand when equality occurs. In this paper we study \ica in relation to \ia from various algebraic and combinatorial perspectives. We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of \ica and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that \ia is the unique minimal prime of \ica and that the embedded primes of \ica are indexed by certain faces of the cone spanned by \A. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to \ia and \ica. The Gr\"obner fan of \ica is shown to refine that of \ia when the codimension of the ideals is at most two.Comment: 25 page

The Slice Algorithm For Irreducible Decomposition of Monomial Ideals

Irreducible decomposition of monomial ideals has an increasing number of applications from biology to pure math. This paper presents the Slice Algorithm for computing irreducible decompositions, Alexander duals and socles of monomial ideals. The paper includes experiments showing good performance in practice.Comment: 25 pages, 8 figures. See http://www.broune.com/ for the data use

Higher Lawrence configurations

Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and face posets of these configurations, and we discuss applications to the statistical theory of log-linear models.Comment: 12 pages. Changes from v1 and v2: minor edits. This version is to appear in the Journal of Combinatorial Theory, Ser.

Maximum information divergence from linear and toric models

We study the problem of maximizing information divergence from a new perspective using logarithmic Voronoi polytopes. We show that for linear models, the maximum is always achieved at the boundary of the probability simplex. For toric models, we present an algorithm that combines the combinatorics of the chamber complex with numerical algebraic geometry. We pay special attention to reducible models and models of maximum likelihood degree one.Comment: 33 pages, 6 figure

The degree of the central curve in semidefinite, linear, and quadratic programming

The Zariski closure of the central path which interior point algorithms track in convex optimization problems such as linear, quadratic, and semidefinite programs is an algebraic curve. The degree of this curve has been studied in relation to the complexity of these interior point algorithms, and for linear programs it was computed by De Loera, Sturmfels, and Vinzant in 2012. We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models. New results from the intersection theory of the space of complete quadrics imply that this is a polynomial in the size of semidefinite matrices with degree equal to the number of constraints. Besides its degree we explore the arithmetic genus of the same curve. We also compute the degree of the central curve for generic linear programs with different techniques which extend to the computation of the same degree for generic quadratic programs.Comment: 15 page

Ideals generated by 2-minors, collection of cells and stack polyominoes

In this paper we study ideals generated by quite general sets of 2-minors of an $m \times n$-matrix of indeterminates. The sets of 2-minors are defined by collections of cells and include 2-sided ladders. For convex collections of cells it is shown that the attached ideal of 2-minors is a Cohen--Macaulay prime ideal. Primality is also shown for collections of cells whose connected components are row or column convex. Finally the class group of the ring attached to a stack polyomino and its canonical class is computed, and a classification of the Gorenstein stack polyominoes is given.Comment: 29 pages, 32 figure

Indispensable monomials of toric ideals and Markov bases

Extending the notion of indispensable binomials of a toric ideal, we define indispensable monomials of a toric ideal and establish some of their properties. They are useful for searching indispensable binomials of a toric ideal and for proving the existence or non-existence of a unique minimal system of binomials generators of a toric ideal. Some examples of indispensable monomials from statistical models for contingency tables are given.Comment: 20 pages, 5 figure

Solving the 100 Swiss Francs Problem

Sturmfels offered 100 Swiss Francs in 2005 to a conjecture, which deals with a special case of the maximum likelihood estimation for a latent class model. This paper confirms the conjecture positively