248 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
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
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
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
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
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