76 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
Frobenius Modules and Hodge Asymptotics
We exhibit a direct correspondence between the potential defining the H^{1,1}
small quantum module structure on the cohomology of a Calabi-Yau manifold and
the asymptotic data of the A-model variation of Hodge structure. This is done
in the abstract context of polarized variations of Hodge structure and
Frobenius modules.Comment: Updated bibliography. Final version published in Commun. Math. Phy
Asymptotic Hodge theory and quantum products
Assuming suitable convergence properties for the Gromov-Witten potential of a
Calabi-Yau manifold one may construct a polarized variation of Hodge
structure over the complexified K\"ahler cone of . In this paper we show
that, in the case of fourfolds, there is a correspondence between ``quantum
potentials'' and polarized variations of Hodge structures that degenerate to a
maximally unipotent boundary point. Under this correspondence, the WDVV
equations are seen to be equivalent to the Griffiths' trasversality property of
a variation of Hodge structure.Comment: References and comments added. To appear in "Advances in Algebraic
Geometry Motivated by Physics", Ed. E. Previatto, Contemporary Mathematic
Restriction of A-Discriminants and Dual Defect Toric Varieties
We study the -discriminant of toric varieties. We reduce its computation
to the case of irreducible configurations and describe its behavior under
specialization of some of the variables to zero. We prove a Gale dual
characterization of dual defect toric varieties and deduce from it the
classsification of such varieties in codimension less than or equal to four.
This classification motivates a decomposition theorem which yields a sufficient
condition for a toric variety to be dual defect. For codimension less than or
equal to four, this condition is also necessary and we expect this to be the
case in general.Comment: 22 pages; In addition to minor corrections, Section 5 has been
expanded and rewritten to include a Gale dual characterization of dual defect
toric varietie
The A-hypergeometric System Associated with a Monomial Curve
We make a detailed analysis of the A-hypergeometric system (or GKZ system)
associated with a monomial curve and integral, hence resonant, exponents. We
characterize the Laurent polynomial solutions and show that these are the only
rational solutions. 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 associated with A. We determine the
holonomic rank r for all integral exponents and show that it is constantly
equal to the degree d of X if and only if X is arithmetically Cohen-Macaulay.
Otherwise there is at least one exponent for which r = d + 1.Comment: Plain Tex, 29 pages. Revised version to appear in Duke Math. J.
Several new results have been adde
On the Locus of Hodge Classes
Let be a family of non singular projective varieties
parametrized by a complex algebraic variety . Fix , an integer ,
and a class of Hodge type . We show that
the locus, on , where remains of type is algebraic. This result,
which in the geometric case would follow from the rational Hodge conjecture, is
obtained in the setting of variations of Hodge structures.Comment: 25 pages, Plain Te
Residues and Resultants
Resultants, Jacobians and residues are basic invariants of multivariate
polynomial systems. We examine their interrelations in the context of toric
geometry. The global residue in the torus, studied by Khovanskii, is the sum
over local Grothendieck residues at the zeros of Laurent polynomials in
variables. Cox introduced the related notion of the toric residue relative to
divisors on an -dimensional toric variety. We establish denominator
formulas in terms of sparse resultants for both the toric residue and the
global residue in the torus. A byproduct is a determinantal formula for
resultants based on Jacobians.Comment: Plain TeX, 22 page
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