Ecalle's paralogarithmic resurgence monomials and effective synthesis

TL;DR

This paper explores Ecalle's paralogarithmic resurgence monomials, a generalization of hyperlogarithms, and demonstrates their effective use in synthesizing analytic vector fields with saddle-node singularities.

Contribution

It introduces the paralogarithmic family of resurgence monomials and applies them to solve the synthesis problem for saddle-node singularities effectively.

Findings

Introduction of paralogarithmic resurgence monomials

Effective synthesis method for saddle-node singularities

Extension of Ecalle's formalism to new special functions

Abstract

Paralogarithms constitute a family of special functions, which are some generalizations of hyperlogarithms. They have been introduced by Jean Ecalle in the context of the classification of complex analytic dynamical systems with irregular singularities, to solve the so-called "synthesis problem" in an effective and very general way. We describe the formalism of resurgence monomials, introduce the paralogarithmic family and present the effective synthesis with paralogarithmic monomials, of analytic vector fields having a saddle-node singularity, following Ecalle.