35 research outputs found
Bounded decomposition in the Brieskorn lattice and Pfaffian Picard--Fuchs systems for Abelian integrals
We suggest an algorithm for derivation of the Picard--Fuchs system of
Pfaffian equations for Abelian integrals corresponding to semiquasihomogeneous
Hamiltonians. It is based on an effective decomposition of polynomial forms in
the Brieskorn lattice. The construction allows for an explicit upper bound on
the norms of the polynomial coefficients, an important ingredient in studying
zeros of these integrals.Comment: 17 pages in LaTeX2
Redundant Picard-Fuchs system for Abelian integrals
We derive an explicit system of Picard-Fuchs differential equations satisfied
by Abelian integrals of monomial forms and majorize its coefficients. A
peculiar feature of this construction is that the system admitting such
explicit majorants, appears only in dimension approximately two times greater
than the standard Picard-Fuchs system.
The result is used to obtain a partial solution to the tangential Hilbert
16th problem. We establish upper bounds for the number of zeros of arbitrary
Abelian integrals on a positive distance from the critical locus. Under the
additional assumption that the critical values of the Hamiltonian are distant
from each other (after a proper normalization), we were able to majorize the
number of all (real and complex) zeros.
In the second part of the paper an equivariant formulation of the above
problem is discussed and relationships between spread of critical values and
non-homogeneity of uni- and bivariate complex polynomials are studied.Comment: 31 page, LaTeX2e (amsart
On almost duality for Frobenius manifolds
We present a universal construction of almost duality for Frobenius
manifolds. The analytic setup of this construction is described in details for
the case of semisimple Frobenius manifolds. We illustrate the general
considerations by examples from the singularity theory, mirror symmetry, the
theory of Coxeter groups and Shephard groups, from the Seiberg - Witten
duality.Comment: 62 pages, a reference adde