Modules of the Abelian integrals and the Picard–Fuchs systems*

We give a simple proof of an isomorphism between two C(t)-modules corresponding to bivariate polynomial H with non-degenerate highest homogeneous part: the module of relative cohomologies 2/dH ∧ 1 and the module of Abelian integrals. Using this isomorphism, we prove the existence and deduce some properties of the corresponding Picard–Fuchs system. Mathematics Subject Classification: 14D05, 32S40, 34C08 The Abelian integral is a result of integration of a polynomial 1-form along a cycle lying on level curve (possibly complex) of a bivarsiate polynomial considered as a function (possibly multivalued) of the value of the polynomial. The Abelian integrals appear naturally when studying bifurcations of limit cycles of planar polynomial vector fields. In particular, zeros of the Abelian integrals are related to limit cycles appearing in polynomial perturbations of polynomial Hamiltonian vector fields. This is the reason why sometimes the question about the number of zeros of the Abelian integrals is sometimes called the infinitesimal Hilbert 16th problem. The traditional approach to the investigation of the Abelian integrals uses properties of the system of linear ordinary differential equations satisfied by the Abelian integrals, the so-called Picard–Fuchs system. This approach is used both in the fundamental general finiteness result of [24, 13] and in exact estimates in the cases of low degree, as in [9]. The existence of such a system can be easily proven due to the very basic properties of branching of the Abelian integrals (see [1]) and was already well known to Riemann if not Gauss. Nevertheless, an effective computation of this system turns out to be a difficult problem. One particular case of this problem (namely of the hyperelliptic integrals) is quite classic (see e.g. [21, 19, 7]). In [18] a generalization of this approach for regular at infinity (see below for exact definition) polynomials in two variables is suggested (in fact, it can be easily generalized for any number * The research was supported by the Killam grant of P Milman and by the James S McDonnell Foundation. 0951-7715/02/051435+10$30.00 © 2002 IOP Publishing Ltd and LMS Publishing Ltd Printed in the UK 1435