128 research outputs found
Higher order selfdual toric varieties
The notion of higher order dual varieties of a projective variety, introduced
in \cite{P83}, is a natural generalization of the classical notion of
projective duality. In this paper we present geometric and combinatorial
characterizations of those equivariant projective toric embeddings that satisfy
higher order selfduality. We also give several examples and general
constructions. In particular, we highlight the relation with Cayley-Bacharach
questions and with Cayley configurations.Comment: 21 page
Explicit formulas for the multivariate resultant
We present formulas for the homogenous multivariate resultant as a quotient
of two determinants. They extend 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.Comment: 30 pages, Late
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
Codimension Theorems for Complete Toric Varieties
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 don't
vanish simultaneously on X.Comment: Small changes including improved introduction. To appear: Proc. AM
Descartes' Rule of Signs for Polynomial Systems supported on Circuits
We give a multivariate version of Descartes' rule of signs to bound the
number of positive real roots of a system of polynomial equations in n
variables with n+2 monomials, in terms of the sign variation of a sequence
associated both to the exponent vectors and the given coefficients. We show
that our bound is sharp and is related to the signature of the circuit.Comment: 25 pages, 3 figure
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.Comment: 12 page
Bases in the solution space of the Mellin system
Local holomorphic solutions z=z(a) to a univariate sparse polynomial equation
p(z) =0, in terms of its vector of complex coefficients a, are classically
known to satisfy holonomic systems of linear partial differential equations
with polynomial coefficients. In this paper we investigate one of such systems
of differential equations which was introduced by Mellin. We compute the
holonomic rank of the Mellin system as well as the dimension of the space of
its algebraic solutions. Moreover, we construct explicit bases of solutions in
terms of the roots of p and their logarithms. We show that the monodromy of the
Mellin system is always reducible and give some factorization results in the
univariate case
Elimination Theory in Codimension Two
New formulas are given for Chow forms, discriminants and resultants arising
from (not necessarily normal) toric varieties of codimension 2. Exact
descriptions are also given for the secondary polygon and for the Newton
polygon of the discriminant.Comment: 20 pages, Late
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