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Redundant Picard-Fuchs system for abelian integrals. (English) Zbl 1011.37042
The authors find an explicit system of Picard-Fuchs differential equations satisfied by abelian integrals of monomial forms and give bounds for its coefficients. These results allow the authors to prove the existence of upper bounds for the number of zeros of abelian integrals on a positive distance from the critical locus. As a consequence they give a partial solution to the tangential Hilbert 16th problem. The second part of the paper is devoted to discuss an equivariant formulation of the problem.

MSC:
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14H70 Relationships between algebraic curves and integrable systems
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
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