Existence of non-algebraic singularities of differential equation

Abstract

An algebraizable singularity is a germ of a singular holomorphic foliation which can be defined in some appropriate local chart by a differential equation with algebraic coefficients. We show that there exists at least countably many saddle-node singularities of the complex plane that are not algebraizable.Comment: 11 page

Extension of germs of holomorphic foliations

“…We follow the procedure exposed in [5]. We have to prove that the image of an embedding ξ : D l −→ (Inv, d) leaves a trace in Inv ∩ L 1 {t} which has empty interior in the topology defined by d 1 .…”

“…In their paper [5], Genzmer and Teyssier introduced a tool that allows to treat this problem from an analytic point of view. Roughly speaking the idea is: given the family L, a surjective map ψ from the set [L] of equivalence clases in L to a space I of invariants is defined; it is assumed that there are appropriate topologies to turn ψ into an "analytic" map.…”

Classification of absolutely dicritical foliations of cusp type

“…Indeed, the action of the group of conjugacies act transversaly to the transverse structures σ and to the moduli of Mattei α. The family As a consequence of the above description of the moduli space of absolutely dicritical foliations, we should be able to prove the existence of a non algebrizable absolutely dicritical foliation using technics developped in [6].…”

An example of a non-algebraizable singularity of a holomorphic foliation

“…We are interested in understanding whether there exists or not an algebraic surface S and a point p on it such that F is holomorphically conjugate to the germ at p of an algebraic foliation on S. Those germs for which such an algebraic foliation exists are called algebraizable. The existence of non-algebraizable singularities remained unknown until Genzmer and Teyssier proved in [GT10] the existence of countably many classes of saddle-node singularities which are not algebraizable. Their proof however, does not provide us with any concrete examples of such singularities and, as far as the author knows, no other examples of non-algebraizable singularities are known.…”