Alicia Dickenstein

Toric dynamical systems

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.

The membership problem for unmixed polynomial ideals is solvable in single exponential time

Abstract Deciding membership for polynomial ideals represents a classical problem of computational commutative algebra which is exponential space hard. This means that the usual algorithms for the membership problem which are based on linear algebra techniques have doubly exponential sequential worst case complexity. We show that the membership problem has single exponential sequential and polynomial parallel complexity for unmixed ideals. More specific complexity results are given for the special cases of zero-dimensional and complete intersection ideals.

Chemical Reaction Systems with Toric Steady States

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.

Explicit formulas for the multivariate resultant

Abstract We present formulas for the multivariate resultant as a quotient of two determinants. They extend the classical Macaulay formulas, and involve matrices of considerably smaller size, whose non-zero entries include coefficients of the given polynomials and coefficients of their Bezoutian. These formulas can also be viewed as an explicit computation of the morphisms and the determinant of a resultant complex.

Sign Conditions for Injectivity of Generalized Polynomial Maps with Applications to Chemical Reaction Networks and Real Algebraic Geometry

We give necessary and sufficient conditions in terms of sign vectors for the injectivity of families of polynomial maps with arbitrary real exponents defined on the positive orthant. Our work relates and extends existing injectivity conditions expressed in terms of Jacobian matrices and determinants. In the context of chemical reaction networks with power-law kinetics, our results can be used to preclude as well as to guarantee multiple positive steady states. In the context of real algebraic geometry, our work recognizes a prior result of Craciun, Garcia-Puente, and Sottile, together with work of two of the authors, as the first partial multivariate generalization of the classical Descartes’ rule, which bounds the number of positive real roots of a univariate real polynomial in terms of the number of sign variations of its coefficients.

Residues in toric varieties

We study residues on a complete toric variety X, which are defined in terms of the homogeneous coordinate ring of X.We first prove a global transformation law for toric residues. When the fan of the toric variety has a simplicial cone of maximal dimension, we can produce an element with toric residue equal to 1. We also show that in certain situations, the toric residue is an isomorphism on an appropriate graded piece of the quotient ring. When X is simplicial, we prove that the toric residue is a sum of local residues. In the case of equal degrees, we also show how to represent X as a quotient (Y\{0})/C* such that the toric residue becomes the local residue at 0 in Y.

Multihomogeneous resultant formulae by means of complexes

The first step in the generalization of the classical theory of homogeneous equations to the case of arbitrary support is to consider algebraic systems with multihomogeneous structure. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bezout types, and including matrices of hybrid type of these two. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman (J. Algebra, 163 (1994) 115; J. Algebraic Geom., 3 (1994) 569). One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. Whenever such formulae exist, we specify the underlying complexes so as to make the resultant matrix explicit. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bezout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with examples showing the kinds of matrices that may be encountered, and illustrations of our MAPLE implementation.

Computing multidimensional residues

Given n polynomials in n variables with a finite number of complex roots, for any of their roots there is a local residue operator assigning a complex number to any polynomial. This is an algebraic, but generally not rational, function of the coefficients. On the other hand, the global residue, which is defined as the sum of the local residues over all roots, has invariance properties which guarantee its rational dependence on the coefficients [9], [27]. In this paper we present symbolic algorithms for evaluating that rational function.

The A-hypergeometric System Associated with a Monomial Curve

Introduction. Inthis paper we make a detailed analysis of the -hypergeometric system (or GKZ system) associated with a monomial curve and integral, hence resonant, exponents. We describe all rational solutions and show in Theorem 1.10 that they are, in fact, Laurent polynomials. We also show that for any exponent there are at most two linearly independent Laurent solutions and that the upper bound is reached if and only if the curve is not arithmetically Cohen-Macaulay. We then construct, for all integral parameters, a basis of local solutions in terms of the roots of the generic univariate polynomial (0.5) associated with . We also determine in Theorem 3.7 the holonomic rank r(α)for all α ∈ Z 2 and show that d ≤ r(α)≤ d +1, where d is the degree of the curve. Moreover, the value d +1 is attained only for those exponents α for which there are two linearly independent rational solutions, and, therefore, r(α)= d for all α if and only if the curve is arithmetically Cohen-Macaulay. Inorder to place these results intheir appropriate con text, we recall the defin ition

Elimination Theory in Codimension 2

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.