Related papers: On real center singularities of complex vector fie…
We study the center-focus problem for planar polynomial vector fields, which can be viewed as a local version of Hilbert's 16th problem. Based on a Lyapunov function approach, we establish novel results regarding the center-focus…
We state some generalizations of a theorem due to G. Darboux, which originally states that a polynomial vector field in the complex plane exhibits a rational first integral and has all its orbits algebraic provided that it exhibits…
We prove that every trajectory of a polynomial vector field on the complex projective plane accumulates to the singular locus of the vector field. This statement represents a holomorphic version of the Poincare-Bendixson theorem and solves…
We study holomorphic vector fields on isolated hypersurface singularities and derive global obstructions to the existence of holomorphic vector fields on compact singular varieties. For a hypersurface germ $(V,0)$ with an isolated…
We analyze the structure of the Poincar\'e map $\Pi$ associated to a monodromic singularity of an analytic family of planar vector fields. We work under two assumptions. The first one is that the family possesses an inverse integrating…
In this paper we give a geometric description of the foliation of a generic real analytic family unfolding a real analytic vector field with a weak focus at the origin, and show that two such families are orbitally analytically equivalent…
We study holomorphic foliations tangent to singular real-analytic Levi-flat hypersurfaces in compact complex manifolds of complex dimension two. We give some hypotheses to guarantee the existence of dicritical singularities of these…
It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}.…
We give essentially unique ``normal forms'' for germs of a holomorphic vector field of the complex plane in the neighborhood of an isolated singularity which is a p:q resonant-saddle. Hence each vector field of that type is conjugate, by a…
In this paper we study germs of holomorphic foliations, at the origin of the complex plane, tangent to Pfaffian hypersurfaces - integral hypersurfaces of real analytic 1-forms - satisfying the Rolle-Khovanskii condition. This hypothesis…
This article studies germs of holomorphic vector fields at the origin of C3 that are tangent to holomorphic foliations of codimension one. Two situations are considered. First, we assume hypotheses on the reduction of singularities of the…
When a singular point of a vector field passes through resonance, a formal invariant cone appears. In the seventies, Pyartli proved that for $(-1,1)$-resonance the cone is in fact analytic and is the degeneration of a family of invariant…
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally…
This paper is about the integrability of complex vector fields in dimension three in a neighborhood of a singular point. More precisely, we study the existence of holomorphic first integrals for isolated singularities of holomorphic vector…
We study analytic deformations of holomorphic differential 1-forms. The initial 1-form is exact homogeneous and the deformation is by polynomial integrable 1-forms. We investigate under which conditions the elements of the deformation are…
We study algebraic integrability of complex planar polynomial vector fields $X=A (x,y)(\partial/\partial x) + B(x,y) (\partial/\partial y) $ through extensions to Hirzebruch surfaces. Using these extensions, each vector field $X$ determines…
We provide sharp lower bounds for the multiplicity of a local holomorphic foliation defined in a complex surface in terms of data associated to a germ of invariant curve. Then we apply our methods to invariant curves whose branches are…
We determine topological and algebraic conditions for a germ of holomorphic foliation $\mathcal F(X)$ induced by a generic vector field $X$ on $(\mathbb{C}^{3},0)$ to have a holomorphic first integral, i.e., a germ of holomorphic map $F…
We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree bounded by a function of the degree of the foliation. We…
We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a…