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Alicia Dickenstein

Publications and source records attributed to Alicia Dickenstein.

At least 37 records · Page 2Linked to original sources
arXiv2015

Implicitization of rational hypersurfaces via linear syzygies: a practical overview

We unveil in concrete terms the general machinery of the syzygy-based algorithms for the implicitization of rational surfaces in terms of the monomials in the polynomials defining the parametrization, following and expanding our joint article with M. Dohm. These algebraic techniques, based on the theory of approximation complexes due to J. Herzog, A, Simis and W. Vasconcelos, were introduced for the implicitization problem by J.-P. Jouanolou, L. Busé, and M. Chardin. Their work was inspired by the practical method of moving curves, proposed by T. Sederberg and F. Chen, translated into the language of syzygies by D. Cox. Our aim is to express the theoretical results and resulting algorithms into very concrete terms, avoiding the use of the advanced homological commutative algebra tools which are needed for their proofs.

math.AG↗
arXiv2014

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.

math.AG↗
arXiv2014

Higher order duality and toric embeddings

The notion of higher order dual varieties of a projective variety is a natural generalization of the classical notion of projective duality, introduced by Piene in 1983. In this paper we study higher order dual varieties of projective toric embeddings. We compute the degree of the second dual variety of a smooth toric threefold in geometric and combinatorial terms, and we classify smooth 2-jet spanned projective embeddings of smooth threefolds whose second dual variety has dimension less than expected. We also describe the tropicalization of the k-th dual variety of an equivariantly embedded (not necessarily normal) toric variety.

math.AG↗
arXiv2014

Implicit dose-response curves

We develop tools from computational algebraic geometry for the study of steady state features of autonomous polynomial dynamical systems via elimination of variables. In particular, we obtain nontrivial bounds for the steady state concentration of a given species in biochemical reaction networks with mass-action kinetics. This species is understood as the output of the network and we thus bound the maximal response of the system. The improved bounds give smaller starting boxes to launch numerical methods. We apply our results to the sequential enzymatic network studied in Markevich et al.(2004) to find nontrivial upper bounds for the different substrate concentrations at steady state. Our approach does not require any simulation, analytical expression to describe the output in terms of the input, or the absence of multistationarity. Instead, we show how to extract information from effectively computable implicit dose-response curves with the use of resultants and discriminants. We moreover illustrate in the application to an enzymatic network, the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteresis diagram provided by a numerical solver. The setting and tools we propose could yield many other results adapted to any autonomous polynomial dynamical system, beyond those where it is possible to get explicit expressions.

q-bio.QM↗
arXiv2013

Plane mixed discriminants and toric jacobians

Polynomial algebra offers a standard approach to handle several problems in geometric modeling. A key tool is the discriminant of a univariate polynomial, or of a well-constrained system of polynomial equations, which expresses the existence of a multiple root. We concentrate on bivariate polynomials and establish an original formula that relates the mixed discriminant of two bivariate Laurent polynomials with fixed support, with the sparse resultant of these polynomials and their toric Jacobian. This allows us to obtain a new proof for the bidegree of the mixed discriminant as well as to establish multipicativity formulas arising when one polynomial can be factored.

math.AG↗
arXiv2012

Nilsson solutions for irregular A-hypergeometric systems

We study the solutions of irregular A-hypergeometric systems that are constructed from Gröbner degenerations with respect to generic positive weight vectors. These are formal logarithmic Puiseux series that belong to explicitly described Nilsson rings, and are therefore called (formal) Nilsson series. When the weight vector is a perturbation of (1,...,1), these series converge and provide a basis for the (multivalued) holomorphic hypergeometric functions in a specific open subset of complex n-space. Our results are more explicit when the parameters are generic or when the solutions studied are logarithm-free. We also give an alternative proof of a result of Schulze and Walther that inhomogeneous A-hypergeometric systems have irregular singularities.

math.AG↗
arXiv2011

Mixed Discriminants

The mixed discriminant of n Laurent polynomials in n variables is the irreducible polynomial in the coefficients which vanishes whenever two of the roots coincide. The Cayley trick expresses the mixed discriminant as an A-discriminant. We show that the degree of the mixed discriminant is a piecewise linear function in the Plucker coordinates of a mixed Grassmannian. An explicit degree formula is given for the case of plane curves.

math.AG↗
arXiv2011

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 parametrized (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.

math.DS↗
arXiv2011

Independent Sets from an Algebraic Perspective

In this paper, we study the basic problem of counting independent sets in a graph and, in particular, the problem of counting antichains in a finite poset, from an algebraic perspective. We show that neither independence polynomials of bipartite Cohen-Macaulay graphs nor Hilbert series of initial ideals of radical zero-dimensional complete intersections ideals, can be evaluated in polynomial time, unless #P=P. Moreover, we present a family of radical zero-dimensional complete intersection ideals J_P associated to a finite poset P, for which we describe a universal Gröbner basis. This implies that the bottleneck in computing the dimension of the quotient by J_P (that is, the number of zeros of J_P) using Gröbner methods lies in the description of the standard monomials.

math.AC↗
arXiv2011

Singular tropical hypersurfaces

We study the notion of singular tropical hypersurfaces of any dimension. We characterize the singular points in terms of tropical Euler derivatives and we give an algorithm to compute all singular points. We also describe non-transversal intersection points of planar tropical curves.

math.AG↗
arXiv2011

Self-dual projective toric varieties

Let T be a torus over an algebraically closed field k of characteristic 0, and consider a projective T-module P(V). We determine when a projective toric subvariety X of P(V) is self-dual, in terms of the configuration of weights of V.

math.AG↗
arXiv2010

A Simple Combinatorial Criterion for Projective Toric Manifolds with Dual Defect

We show that any smooth lattice polytope P with codegree greater or equal than (dim(P)+3)/2 (or equivalently, with degree smaller than dim(P)/2), defines a dual defective projective toric manifold. This implies that P is Q-normal (in the terminology of a recent paper by Di Rocco, Piene and the first author) and answers partially an adjunction-theoretic conjecture by Beltrametti and Sommese. Also, it follows that smooth lattice polytopes with this property are precisely strict Cayley polytopes, which completes the answer of a question of Batyrev and the second author in the nonsingular case.

math.CO↗
arXiv2010

How far is complex balancing from detailed balancing?

The aim of this article is to build on the use of tools from computational algebra initiated in Craciun, Dickenstein, Shiu, Sturmfels (JSC, 2009), for the study of general kinetic systems, which have a wide range of applications in chemistry and biology. We clarify the relation between the algebraic conditions that must be satisfied by the reaction constants in general (mass action) kinetics systems for the existence of detailed or complex balancing equilibria. The main properties of these systems have been set by Horn, Jackson and Feinberg.

math.DS↗
arXiv2009

The structure of bivariate rational hypergeometric functions

We describe the structure of all codimension-two lattice configurations $A$ which admit a stable rational $A$-hypergeometric function, that is a rational function $F$ all whose partial derivatives are non zero, and which is a solution of the $A$-hypergeometric system of partial differential equations defined by Gel'fand, Kapranov and Zelevinsky. We show, moreover, that all stable rational $A$-hypergeometric functions may be described by toric residues and apply our results to study the rationality of bivariate series whose coefficients are quotients of factorials of linear forms.

math.AG↗
arXiv2009

Additive Edge Labelings

Let G=(V,E) be a graph and d a positive integer. We study the following problem: for which labelings f_E: E \to Z_d is there a labeling f_V:V \to Z_d such that f_E(i,j) = f_V(i) + f_V(j) (mod d), for every edge (i,j) in 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.

math.CO↗
arXiv2009

Classifying smooth lattice polytopes via toric fibrations

We define Q-normal lattice polytopes. Natural examples of such polytopes are Cayley sums of strictly combinatorially equivalent lattice polytopes, which correspond to particularly nice toric fibrations, namely toric projective bundles. In a recent paper Batyrev and Nill have suggested that there should be a bound, N(d), such that every lattice polytope of degree d and dimension at least N(d) decomposes as a Cayley sum. We give a sharp answer to this question for smooth Q-normal polytopes. We show that any smooth Q-normal lattice polytope P of dimension n and degree d is a Cayley sum of strictly combinatorially equivalent polytopes if n is greater than or equal to 2d+1. The proof relies on the study of the nef value morphism associated to the corresponding toric embedding.

math.AG↗
arXiv2008

Matrix representations for toric parametrizations

In this paper we show that a surface in P^3 parametrized over a 2-dimensional toric variety T can be represented by a matrix of linear syzygies if the base points are finite in number and form locally a complete intersection. This constitutes a direct generalization of the corresponding result over P^2 established in [BJ03] and [BC05]. Exploiting the sparse structure of the parametrization, we obtain significantly smaller matrices than in the homogeneous case and the method becomes applicable to parametrizations for which it previously failed. We also treat the important case T = P^1 x P^1 in detail and give numerous examples.

math.AG↗