Redundant Picard-Fuchs system for Abelian integrals

TL;DR

This paper develops an explicit, larger Picard-Fuchs system for Abelian integrals, enabling bounds on the zeros of these integrals and contributing to the tangential Hilbert 16th problem, with applications to polynomial critical values.

Contribution

It introduces a novel, explicit Picard-Fuchs system that is larger but more manageable, providing new bounds on Abelian integral zeros and insights into polynomial critical value distribution.

Findings

Derived explicit Picard-Fuchs system with majorants

Established bounds for zeros of Abelian integrals

Linked critical value spread to polynomial non-homogeneity

Abstract

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.