Logarithmic models and meromorphic functions in dimension two

TL;DR

This paper constructs logarithmic models in real and complex settings to produce meromorphic functions and vector fields with specified singularity, zero, and pole structures, using geometric and foliation data.

Contribution

It introduces a method to build logarithmic models that realize prescribed indeterminacy, zeroes, poles, and sectorial decompositions in meromorphic functions and vector fields.

Findings

Constructed meromorphic functions with specified indeterminacy structures.

Built vector fields with prescribed sectorial decompositions.

Provided a geometric framework for modeling singularities and foliations.

Abstract

In this article we describe the construction of logarithmic models in both real and complex cases. A logarithmic model is a germ of closed meromorphic 1-form with simple poles - and the analytic foliation defined by it - produced upon some specified geometric data: the structure of dicritical (non-invariant) components in the exceptional divisor of its reduction of singularities, a prescribed finite set of separatrices - invariant analytic branches at the origin - and Camacho-Sad indices with respect to these separatrices. As an application, we use logarithmic models in order to construct real and complex germs of meromorphic functions with a given indeterminacy structure and prescribed sets of zeroes and poles. Also, in the real case, in the specific case where all trajectories accumulating at the origin are contained in analytic curves, logarithmic models are used in order to build germs of analytic vector fields with a given Bendixson's sectorial decomposition of a neighborhood of $0 \in \R^{2}$ into hyperbolic, parabolic and elliptic sectors. As a consequence, we can produce real meromorphic functions with prescribed sectorial decompositions.