What question did this study set out to answer?

The aim is to classify germs of foliations induced by real-analytic vector fields with singularities using real-analytic invariants.

May 16, 2026Open Access

Analytic Invariants for Real-Analytic Germs of Vector Fields in (R^2, 0)

Key Points

The aim is to classify germs of foliations induced by real-analytic vector fields with singularities using real-analytic invariants.
Classifying real-analytic vector fields in R², specifically those having singularities of order n where n≥2.
Introducing real-analytic Thom’s invariants for classification in a specific subclass of vector fields.
Addressing the realization problem by constructing vector fields based on prescribed real-analytic invariants and constants.
Real-analytic Thom’s invariants successfully provide a means for classifying vector fields in the chosen subclass.
Construction of vector fields with desired invariants demonstrates viability of the realization problem.

Abstract

This article deals with two problems related with germs of foliations induced by real–analytic vector fields in (R ², 0) having singularity at 0 of order n, n 2. We will denote this class of vector fields by Aₙ. The goal of the first problem is to obtain a set of real–analytic invariants, called real–analytic Thom’s invariants, that allow us to achieve the real–analytic classification of vector fields satisfying genericity assumptions in a subclass A⁰ₙ of Aₙ. The second problem we deal with is called the realization problem, which consists in constructing a real analytic vector field having as real–analytic invariants a suitable group G of conformal mappings and a collection \ ₈₉\ of (n-3) (n-2) /2 real constants that are given in advance.

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