On exponentials of exponential generating series

TL;DR

This paper explores the exponential map on exponential generating series, establishing a bijection between rational or algebraic series and their exponential counterparts over certain fields.

Contribution

It proves that the exponential and logarithm maps induce bijections between rational/algebraic series and their exponential forms in specific algebraic settings.

Findings

The exponential map ${ m exp}_!$ is a bijection between rational series and their exponential forms.

The inverse map ${ m log}_!$ also induces a bijection for algebraic series.

Results hold over fields subfield of algebraically closed fields of characteristic p.

Abstract

Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\mathop{exp}}_!:X\mathbb K[[X]]\longrightarrow 1+X\mathbb K[[X]]$ for the associated Lie group formed by exponential generating series with constant coefficient 1 over an arbitrary field $\mathbb K$. The main result of this paper states that the map ${\mathop{exp}}_!$ (and its inverse map ${\mathop{log}}_!$) induces a bijection between rational, respectively algebraic, series in $X\mathbb K [[X]]$ and $1+X\mathbb K[[X]]$ if the field $\mathbb K$ is a subfield of the algebraically closed field $\bar{\mathbb F}_p$ of characteristic $p$.