TY - JOUR AB - Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses of symmetric matrices satisfying linear constraints. This class includes Gaussian graphical models. We develop a general theory of exponential varieties. These are derived from hyperbolic polynomials and their integral representations. We compare the multidegrees and ML degrees of the gradient map for hyperbolic polynomials. AU - Michałek, Mateusz AU - Sturmfels, Bernd AU - Uhler, Caroline AU - Zwiernik, Piotr ID - 1480 IS - 1 JF - Proceedings of the London Mathematical Society TI - Exponential varieties VL - 112 ER - TY - JOUR AB - Relational models for contingency tables are generalizations of log-linear models, allowing effects associated with arbitrary subsets of cells in the table, and not necessarily containing the overall effect, that is, a common parameter in every cell. Similarly to log-linear models, relational models can be extended to non-negative distributions, but the extension requires more complex methods. An extended relational model is defined as an algebraic variety, and it turns out to be the closure of the original model with respect to the Bregman divergence. In the extended relational model, the MLE of the cell parameters always exists and is unique, but some of its properties may be different from those of the MLE under log-linear models. The MLE can be computed using a generalized iterative scaling procedure based on Bregman projections. AU - Klimova, Anna AU - Rudas, Tamás ID - 1833 JF - Journal of Multivariate Analysis TI - On the closure of relational models VL - 143 ER - TY - JOUR AB - Let G be a graph on the vertex set V(G) = {x1,…,xn} with the edge set E(G), and let R = K[x1,…, xn] be the polynomial ring over a field K. Two monomial ideals are associated to G, the edge ideal I(G) generated by all monomials xixj with {xi,xj} ∈ E(G), and the vertex cover ideal IG generated by monomials ∏xi∈Cxi for all minimal vertex covers C of G. A minimal vertex cover of G is a subset C ⊂ V(G) such that each edge has at least one vertex in C and no proper subset of C has the same property. Indeed, the vertex cover ideal of G is the Alexander dual of the edge ideal of G. In this paper, for an unmixed bipartite graph G we consider the lattice of vertex covers LG and we explicitly describe the minimal free resolution of the ideal associated to LG which is exactly the vertex cover ideal of G. Then we compute depth, projective dimension, regularity and extremal Betti numbers of R/I(G) in terms of the associated lattice. AU - Mohammadi, Fatemeh AU - Moradi, Somayeh ID - 1547 IS - 3 JF - Bulletin of the Korean Mathematical Society TI - Resolution of unmixed bipartite graphs VL - 52 ER - TY - JOUR AB - We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality. AU - Brooks, Christopher AU - Martin Del Campo Sanchez, Abraham AU - Sottile, Frank ID - 1579 IS - 6 JF - Transactions of the American Mathematical Society TI - Galois groups of Schubert problems of lines are at least alternating VL - 367 ER - TY - JOUR AB - We prove that the three-state toric homogeneous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals is generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that they are generated by degree six binomials. AU - Noren, Patrik ID - 1997 IS - May-June JF - Journal of Symbolic Computation TI - The three-state toric homogeneous Markov chain model has Markov degree two VL - 68/Part 2 ER -