88 research outputs found
Counting Solutions to Binomial Complete Intersections
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.Comment: Several minor improvements. Final version to appear in the J. of
Complexit
Multihomogeneous resultant formulae by means of complexes
We provide conditions and algorithmic tools so as to classify and construct
the smallest possible determinantal formulae for multihomogeneous resultants
arising from Weyman complexes associated to line bundles in products of
projective spaces. We also examine the smallest Sylvester-type matrices,
generically of full rank, which yield a multiple of the resultant. We
characterize the systems that admit a purely B\'ezout-type matrix and show a
bijection of such matrices with the permutations of the variable groups. We
conclude with examples showing the hybrid matrices that may be encountered, and
illustrations of our Maple implementation. Our approach makes heavy use of the
combinatorics of multihomogeneous systems, inspired by and generalizing results
by Sturmfels-Zelevinsky, and Weyman-Zelevinsky.Comment: 30 pages. To appear: Journal of Symbolic Computatio
Integrating Singular Functions on the Sphere
We obtain rigorous results concerning the evaluation of integrals on the two
sphere using complex methods. It is shown that for regular as well as singular
functions which admit poles, the integral can be reduced to the calculation of
residues through a limiting procedure.Comment: 15 pages, revte
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.Comment: 14 page
Complex-linear invariants of biochemical networks
The nonlinearities found in molecular networks usually prevent mathematical
analysis of network behaviour, which has largely been studied by numerical
simulation. This can lead to difficult problems of parameter determination.
However, molecular networks give rise, through mass-action kinetics, to
polynomial dynamical systems, whose steady states are zeros of a set of
polynomial equations. These equations may be analysed by algebraic methods, in
which parameters are treated as symbolic expressions whose numerical values do
not have to be known in advance. For instance, an "invariant" of a network is a
polynomial expression on selected state variables that vanishes in any steady
state. Invariants have been found that encode key network properties and that
discriminate between different network structures. Although invariants may be
calculated by computational algebraic methods, such as Gr\"obner bases, these
become computationally infeasible for biologically realistic networks. Here, we
exploit Chemical Reaction Network Theory (CRNT) to develop an efficient
procedure for calculating invariants that are linear combinations of
"complexes", or the monomials coming from mass action. We show how this
procedure can be used in proving earlier results of Horn and Jackson and of
Shinar and Feinberg for networks of deficiency at most one. We then apply our
method to enzyme bifunctionality, including the bacterial EnvZ/OmpR osmolarity
regulator and the mammalian
6-phosphofructo-2-kinase/fructose-2,6-bisphosphatase glycolytic regulator,
whose networks have deficiencies up to four. We show that bifunctionality leads
to different forms of concentration control that are robust to changes in
initial conditions or total amounts. Finally, we outline a systematic procedure
for using complex-linear invariants to analyse molecular networks of any
deficiency.Comment: 36 pages, 6 figure
Linear Toric Fibrations
These notes are based on three lectures given at the 2013 CIME/CIRM summer
school. The purpose of this series of lectures is to introduce the notion of a
toric fibration and to give its geometrical and combinatorial
characterizations. Polarized toric varieties which are birationally equivalent
to projective toric bundles are associated to a class of polytopes called
Cayley polytopes. Their geometry and combinatorics have a fruitful interplay
leading to fundamental insight in both directions. These notes will illustrate
geometrical phenomena, in algebraic geometry and neighboring fields, which are
characterized by a Cayley structure. Examples are projective duality of toric
varieties and polyhedral adjunction theory
Tropical surface singularities
In this paper, we study tropicalisations of singular surfaces in toric
threefolds. We completely classify singular tropical surfaces of
maximal-dimensional type, show that they can generically have only finitely
many singular points, and describe all possible locations of singular points.
More precisely, we show that singular points must be either vertices, or
generalized midpoints and baricenters of certain faces of singular tropical
surfaces, and, in some cases, there may be additional metric restrictions to
faces of singular tropical surfaces.Comment: A gap in the classification was closed. 37 pages, 29 figure
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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
Rational hypergeometric functions
Multivariate hypergeometric functions associated with toric varieties were introduced by Gel\u27fand, Kapranov and Zelevinsky. Singularities of such functions are discriminants, that is, divisors projectively dual to torus orbit closures. We show that most of these potential denominators never appear in rational hypergeometric functions. We conjecture that the denominator of any rational hypergeometric function is a product of resultants, that is, a product of special discriminants arising from Cayley configurations. This conjecture is proved for toric hypersurfaces and for toric varieties of dimension at most three. Toric residues are applied to show that every toric resultant appears in the denominator of some rational hypergeometric function
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
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