Modules of Abelian Integrals and Picard-fuchs Systems

We give a simple proof of an isomorphism between the two C[t]-modules: the module of relative cohomologies Λ/dH ∧ Λ and the module ofAbelian integrals corresponding to a Morse-plus polynomial H in two variables.Using this isomorphism, we prove existence and deduce some properties of thecorresponding Picard-Fuchs system. Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA E-mail: [email protected] Modules of Abelian integrals and Picard-Fuchs systems2 Abelian integral is a result of integration of a polynomial one-form along a cyclelying on level curve (possibly complex) of a bivariate polynomial considered as afunction (possibly multivalued) of the value of the polynomial. Abelian integralsappear naturally when studying bifurcations of limit cycles of planar polynomial vectorfields. In particular, zeros of Abelian integrals are related to limit cycles appearingin polynomial perturbations of polynomial Hamiltonian vector fields. This is thereason why sometimes the question about the number of zeroes of Abelian integralsis sometimes called infinitesimal Hilbert 16 problem.The traditional approach to the investigation of Abelian integrals uses propertiesof the system of linear ordinary differential equations satisfied by the Abelian integrals,the so-called Picard-Fuchs system. The existence of such a system can be easily provendue to the very basic properties of branching of Abelian integrals, see [1], and waswell known already to Riemann if not Gauss. Nevertheless an effective computationof 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. [18, 16]. In [15] ageneralization 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 anynumber of variables). The main idea of [15] is to trade the minimality of the size ofthe system (thus redundant) for an explicitness of the construction and control oncoefficients. Another, probably not less important, gain is that the resulting systemis not only Fuchsian, but also has a hypergeometric form.The control on the magnitude of coefficients in [15] is very important from theinfinitesimal Hilbert 16 problem point of view. Indeed, recent progress towards itssolution is partly based on an upper estimate on the number of zeros of solution ofdifferential equation in terms of magnitude of the coefficients of the equation, a longforgotten principle [20] rediscovered in [9] (see also [13] for a simple proof). Thoughmore complicated, this principle still holds for polynomial systems of differentialequations, see [13] (polynomiality is essential, see [11]). In slightly modified form, thisprinciple allows to give an upper bound on the number of zeros of an Abelian integralin terms of the minimal distance between critical values of its (regular at infinity)Hamiltonian, see [15]. It also allows to solve the infinitesimal Hilbert problem forhyperelliptic Hamiltonians, see [12].The Picard-Fuchs systems discussed in this paper is irredundant in the sense thatis has the minimal possible dimension (namely the dimension equal to the dimensionof H({H(x, y) = t},C) for a generic t). This minimality allows to guess most of theimportant information about the system if the critical values of the Hamiltonian aredistinct and the Hamiltonian is regular at infinity (so-called Morse-plus Hamiltonians).We prove existence of such system using decomposition in Petrov modules.In [3] L.Gavrilov proved that Abelian integrals corresponding to a generic enoughHamiltonian form a finitely generated free C[t]-module, the so-called Petrov module.The local counterpart of this statement is due to E.Brieskorn and M. Sebastiany. Theproof in [3] contains a gap (more precisely, a vague reference to a deep nondegeneracyresult). This gap was sealed in the recent preprint of Yu.Ilyashenko [8] by astraightforward but somewhat mysterious calculations. Moreover, in [6] the involvedconstant is computed and exact formulae are given. We suggest an elementary proofof this result.The main idea of [15] was to use a connection between division with remainder ofpolynomials and differentiation of Abelian integrals given by Gelfand-Leray formula.In this work we replace the explicit division with remainder by decomposition in Petrov Modules of Abelian integrals and Picard-Fuchs systems3 modules in order to get the same result. This is still enough for the construction ofthe system, though the result is less explicit. Yet one can still guess all singularpoints and get some information about coefficients. However, we show that theresulting irredundant system is not always Fuchsian, namely it can have regular andnot Fuchsian point at infinity. 0.1. Acknowledgments This paper was initially intended to be an Appendix in our joint work withS.Yakovenko [15]. I’m grateful to S. Yakovenko for the numerous discussions andhelp in preparation of this text. I’m grateful to Yulij Ilyashenko for an opportunityto read his recent preprint and numerous discussions. 1. Irredundant systems of Picard–Fuchs equations 1.1. Genericity and generalities In what follow we always assume that our polynomialH(x, y) is Morse-plus, i.e. satisfythe following genericity conditions: (i) Polynomial H(x, y) is regular at infinity, i.e its highest homogeneous part Ĥ(x, y)is a product of pairwise different linear factors, (ii) Polynomial H(x, y) is Morse, i.e. all its critical values are pairwise different. Immediate consequences of regularity at infinity include(i) Ĥ has an isolated critical point (necessarily of multiplicity μ = n) at the origin(x, y) = (0, 0);(ii) the level curve {Ĥ = 1} ⊂ C is nonsingular. (iii) the level curves of H intersect transversally the line at infinity(iv) foliation of C by level curves of H is locally topologically trivial over C\Σ, whereΣ is the set of (degH − 1) critical values of H . By Abelian integrals we mean the result of integration of a one-form ω along acontinuous family of cycles δ(t) ⊂ {H = t} considered as a function of t: