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.

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: 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.