Variance of vector fields -- Definition and properties

TL;DR

This paper provides a comprehensive presentation of the variance of vector fields, including detailed proofs and definitions, highlighting its importance in understanding the structure of resonant vector fields.

Contribution

It offers a self-contained exposition of the variance of vector fields, clarifying and expanding on previous work by Ecalle, Vallet, and Schlomiuk with complete proofs.

Findings

Complete proofs of variance properties

Formulas for the mould of nilpotent parts

Illustration of variance's relevance in resonant vector fields

Abstract

We give a self contained presentation of the notion of variance of a vector field introduced by Jean Ecalle and Bruno Vallet in \cite{ev} following a previous work of Jean Ecalle and Dana Schlomiuk in \cite{es}. We give complete proofs and definitions of various results stated in these articles. Following J. Ecalle and D. Schlomiuk, We illustrate the interest of the variance by giving a complete proof of the formulas for the mould defining the nilpotent part of a resonant vector field.