Searcharxiv⌕ Search

arXiv subjects

Yohann Genzmer

Publications and source records attributed to Yohann Genzmer.

12 recordsLinked to original sources
arXiv2024

On the Saito number of plane curves

In this work we study the \emph{Saito number} of a plane curve and we present a method to determine the minimal Saito number for plane curves in a given equisingularity class, that gives rise to an actual algorithm. In particular situations, we also provide various formulas for this number. In addition, if $ν_0$ and $ν_1$ are two coprime positive integers and $N>0$ then we show that for any $1\leq k\leq \left [\frac{Nν_0}{2}\right ]$ there exits a plane curve equisingular to the curve $$y^{Nν_0}-x^{Nν_1}=0$$ such that its Saito number is precisely $k$.

math.AG↗
arXiv2024

The Saito vector field of a germ of complex plane curve

In this article, we prove that an algorithm introduced by the author in a previous work and giving the generic dimension of the moduli space of a germ of curve in the complex plane that is the union of smooth curves, can be used identically to find this dimension for any kind of germ of plane curve.

math.DS↗
arXiv2023

On the algebraicity of germs of meromorphic functions

In this article we prove that every germ of analytic meromorphic function at $(\mathbb{C}^{2},0)$ is equivalent, under the right composition by a germ of biholomorphism, to a germ of algebraic meromorphic function. An analogous result is also true for real analytic meromorphic functions.

math.CV↗
arXiv2019

On the Saito's basis and the Tjurina Number for Plane Branches

We introduce the concept of good Saito's basis for a plane curve $S$ and we explore it to obtain a formula for the minimal Tjurina number in a topological class. In particular, we present a positive answer for a question of Dimca and Greuel relating the Tjurina number and the Milnor number for a singular irreducible plane curve.

math.AG↗
arXiv2016

Schlesinger foliation for deformations of foliations

In this article, we show that for any deformation of analytic foliations, there exists a maximal analytic singular foliation on the space of parameters along the leaves of which the deformation is integrable.

math.DS↗
arXiv2016

The Poincaré problem in the dicritical case

We develop a study on local polar invariants of planar complex analytic foliations at $(\mathbb{C}^{2},0)$, which leads to the characterization of second type foliations and of generalized curve foliations, as well as a description of the $GSV$-index. We apply it to the Poincaré problem for foliations on the complex projective plane $\mathbb{P}^{2}_{\mathbb{C}}$, establishing, in the dicritical case, conditions for the existence of a bound for the degree of an invariant algebraic curve $S$ in terms of the degree of the foliation $\mathcal{F}$. We characterize the existence of a solution for the Poincaré problem in terms of the structure of the set of local separatrices of $\mathcal{F}$ over the curve $S$. Our method, in particular, recovers the known solution for the non-dicritical case, ${\rm deg}(S) \leq {\rm deg}(\mathcal{F}) + 2$.

math.DS↗
arXiv2012

Classification of absolutely dicritical foliations of cusp type

We give a classification of absolutely dicritical foliations of cusp type, that is, the germ of singularities of complex foliations in the complex plane topologically equivalent to the singularity given by the level of the meromorphic function \frac{y^{2}+x^{3}}{xy}.

math.CV↗
arXiv2010

Existence of non-algebraic singularities of differential equation

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.

math.DS↗
arXiv2009

Normal forms of foliations and curves defined by a function with a generic tangent cone

We first describe the local and global moduli spaces of germs of foliations defined by analytic functions in two variables with p transverse smooth branches, and with integral multiplicities (in the univalued holomorphic case) or complex multiplicities (in the multivalued ''Darboux'' case). We specify normal forms in each class. Then we study on these moduli space the distribution C induced by the following equivalence relation: two points are equivalent if and only if the corresponding foliations have the same analytic invariant curves up to analytical conjugacy. Therefore, the space of leaves of C is the moduli space of curves. We prove that C is rationally integrable. These rational integrals give a complete system of invariants for these generic plane curves, which extend the well-known cross-ratios between branches.

math.CV↗