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Gauss-Manin connection associated to the versal deformation of the singularity \(A_\mu\) and zeros of the hyperelliptic integral. (Connexion de Gauss-Manin associée à la déformation verselle de la singularité \(A_\mu\) et zéros de l’intégrale hyperelliptique.) (French) Zbl 1064.32023

The author investigates systems of differential equations which are satisfied by hyperelliptic integrals. His main very interesting result is an explicit bound for the multiplicity of zeros of Abelian integrals. The relation with the Hilbert 16th problem about limit cycles is presented in the final part of the paper.

MSC:

32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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References:

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