Analytic classification of plane branches up to multiplicity 4

Hefez, Abramo; Hernandes, Marcelo Escudeiro

doi:10.1016/j.jsc.2008.09.003

Cited by 20 publications

(23 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This problem is a particular case of an open problem known as the Zariski problem. It has only a very few satisfying answer: Zariski [16] for the very first treatment of some particular cases, Hefez and Hernandez [5,6] for the irreducible curves, Granger [9] in the homogeneous topological class and [2] for some results which are particular case of our present results. Our strategy that we already introduced in a previous work [8], differs from all this works: from our description of the moduli space M, we consider the distribution C on M induced by the equivalence relation ∼ c : two foliations represented by two points in M are in a same orbit of this distribution if and only if they induce the same curve up to analytic conjugacy.…”

Section: Introductionmentioning

confidence: 81%

“…For the topological type (3,5) with multiplicities (0, 0, n 1 , n 2 , n 3 , n 4 ) considered in Figure (6), since 2 × 3 − 1 × 5 = 1, we obtain that u = 2 and v = 1. We have ν c = 15 × 4 − 5 − 3 = 52.…”

Section: Construction Of the Local Normal Formsmentioning

confidence: 89%

“…Let U be the covering of D by these two open sets U 0 and U ∞ -see figure (5) in Appendix A. We know from the proposition (3.6) of the Appendix A that…”

Section: The Infinitesimal Descriptionmentioning

confidence: 99%

See 2 more Smart Citations

Moduli spaces for topologically quasi-homogeneous functions

Genzmer¹,

Paul²

2016

jsing

View full text Add to dashboard Cite

We consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distribution on this moduli space. By studying the infinitesimal generators of this distribution, we can compute the generic dimension of the moduli space for the curves, and we obtain the corresponding generic normal forms.

show abstract

Section: Introductionmentioning

confidence: 81%

Section: Construction Of the Local Normal Formsmentioning

confidence: 89%

See 1 more Smart Citation

Moduli spaces for topologically quasi-homogeneous functions

Genzmer¹,

Paul²

2016

jsing

View full text Add to dashboard Cite

show abstract

“…The classification of multiplicity 4 and genus 2 branches is given in the table below, extracted from [HH2]. Since gcd(4, v 1 , v 2 ) = 1 and v 2 > 2v 1 , we may write v 1 = 2k 1 and v 2 = 4k 1 + d, where k 1 and d are odd numbers.…”

Section: Multiplicity Four and Genus Twomentioning

confidence: 99%

On Polars of Plane Branches

Hefez

Hernandes

Iglesias

2017

Trends in Mathematics

View full text Add to dashboard Cite

It is well known that the equisingularity class of the general polar of a plane branch is not the same for all branches in a given equisingularity class, but it is the same for sufficiently general ones (cf. [C2]) and depends upon the analytic type of the branch. The aim of this paper is to go beyond generality and show how one could describe the equisingularity classes of (general) polars of all branches in a given equisingularity class, making use of the analytic classification of branches as described in [HH3]. We will show how this works in some particular equisingularity classes for which one has the complete explicit analytic classification, and in particular for all branches of multiplicity less or equal than four, based on the classification given in [HH2].

show abstract

“…As an application of the results presented here we refer to [7], where all branches up to multiplicity four are classified.…”

Section: Passage From the A 1 -Equivalence To The A-equivalencementioning

confidence: 99%

The analytic classification of plane branches

Hefez

Hernandes

2011

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

An effective solution of the problem of analytic classification of plane branches is given. A finite stratification of any given equisingularity class of plane branches is determined and normal forms for each stratum are exhibited in such a way that two branches in normal form are easily recognized to be analytically equivalent or not. In this way, we solve the main problems proposed by O. Zariski (Le problème des modules pour les branches planes.

show abstract

Analytic classification of plane branches up to multiplicity 4

Cited by 20 publications

References 15 publications

Moduli spaces for topologically quasi-homogeneous functions

Moduli spaces for topologically quasi-homogeneous functions

On Polars of Plane Branches

The analytic classification of plane branches

Contact Info

Product

Resources

About