Descripción: Let X be a complete toric variety with homogeneous coordinate ring S. In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of S generated by dim(X) + 1 homogeneous polynomials that do not vanish simultaneously on X. ©2005 American Mathematical Society.

Descripción: Let G = (V, E) be a graph and d a positive integer. We study the following problem: for which labelings fE : E → Zd is there a labeling fV : V → Zd such that fE (i, j) = fV (i) + fV (j) (mod d), for every edge (i, j) ∈ E? We also explore the connections of the equivalent multiplicative version to toric ideals. We derive a polynomial algorithm to answer these questions and to obtain all possible solutions. © 2009 Elsevier B.V. All rights reserved.

Descripción: New formulae are given for Chow forms, discriminants and resultants arising from (not necessarily normal) toric varieties of codimension 2. The Newton polygon of the discriminant is determined exactly. © 2002 Elsevier Science Ltd. All rights reserved.

Descripción: We study the problem of counting the total number of affine solutions of a system of n binomials in n variables over an algebraically closed field of characteristic zero. We show that we may decide in polynomial time if that number is finite. We give a combinatorial formula for computing the total number of affine solutions (with or without multiplicity) from which we deduce that this counting problem is # P-complete. We discuss special cases in which this formula may be computed in polynomial time; in particular, this is true for generic exponent vectors. © 2006 Elsevier Inc. All rights reserved.

Descripción: Fil:Dickenstein, A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

Descripción: We present examples that show that in dimension higher than one or codimension higher than two, there exist toric ideals IA such that no binomial ideal contained in IA and of the same dimension is a complete intersection. This result has important implications in sparse elimination theory and in the study of the Horn system of partial differential equations. © 2006 American Mathematical Society.

Descripción: We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank. © 2004 Elsevier Inc. All rights reserved.

Descripción: Fil:Dickenstein, A. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.

Descripción: We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a strict Cayley polytope if n ≥ 2 d + 1. This gives a sharp answer, for this class of polytopes, to a question raised by V.V. Batyrev and B. Nill. © 2009 Elsevier Inc. All rights reserved.

Descripción: Toric dynamical systems are known as complex balancing mass action systems in the mathematical chemistry literature, where many of their remarkable properties have been established. They include as special cases all deficiency zero systems and all detailed balancing systems. One feature is that the steady state locus of a toric dynamical system is a toric variety, which has a unique point within each invariant polyhedron. We develop the basic theory of toric dynamical systems in the context of computational algebraic geometry and show that the associated moduli space is also a toric variety. It is conjectured that the complex balancing state is a global attractor. We prove this for detailed balancing systems whose invariant polyhedron is two-dimensional and bounded. © 2009 Elsevier Ltd. All rights reserved.