An explicit formula for the determinant of the Abelian integral matrix

TL;DR

This paper derives an explicit formula for the determinant of the Abelian integral matrix associated with a polynomial in two variables, revealing its polynomial nature and critical value zeros.

Contribution

It provides a new explicit formula for the determinant of the Abelian integral matrix, extending Ilyashenko's polynomial result to an exact expression.

Findings

Determinant is a polynomial in t of degree n*n

Zeros of the determinant are the critical values of h

Explicit formula enables direct computation of the determinant

Abstract

We consider a polynomial h(x,y) in two complex variables of degree n+1>1 with a generic higher homogeneous part. The rank of the first homology group of its nonsingular level curve h(x,y)=t is n*n. To each 1- form in the variable space and a generator of the homology group one associates the (Abelian) integral of the form along the generator. The Abelian integral is a multivalued function in t. For a fixed canonic tuple of n*n monomial 1- forms we consider the multivalued square matrix function in t whose elements are the Abelian integrals of the forms along the generators. Its determinant does not depend on the choice of the generators in the homology group (up to change of sign, which corresponds to change of generator system that reverses orientation). In 1999 Yu.S.Ilyashenko proved that the determinant of the Abelian integral matrix is a polynomial in t of degree n*n whose zeroes are the critical values of h. We give an explicit formula for the determinant.