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Center conditions: Rigidity of logarithmic differential equations. (English) Zbl 1049.32033

The author shows that any degree \(d\) deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic also, thus generalizing Ilyashenko’s result on Hamiltonian equations. Some applications are also given. A hint towards studying some of the notions presented here to the case of an algebraically closed field instead of \(\mathbb {C}\) is also presented.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
32S30 Deformations of complex singularities; vanishing cycles
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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