Normal forms of parabolic logarithmic transseries

TL;DR

This paper develops an algorithmic approach to find formal normal forms for parabolic logarithmic transseries, providing explicit formulas and eliminating residual terms in broader classes, advancing the understanding of their structure.

Contribution

It introduces a fixed point theorem-based algorithm for normalizing parabolic logarithmic transseries, improving upon previous transfinite elimination methods.

Findings

Normal forms are obtained via fixed point theorems and limits of Picard sequences.

Explicit formulas for residual coefficients are provided.

Residual terms can be eliminated in larger logarithmic classes.

Abstract

We give formal normal forms for parabolic logarithmic transseries $f=z+\cdots \, $, with respect to parabolic logarithmic normalizations. Normalizations are given algorithmically, using fixed point theorems, as limits of Picard's sequences in appropriate complete metric spaces, in contrast to transfinite \emph{term-by-term} eliminations described in former works. Furthermore, we give the explicit formula for the residual coefficient in the normal form and show that, in the larger logarithmic class, we can even eliminate the residual term from the normal form.