On exponentials of exponential generating series

Abstract

Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\exp }_{!}:X\mathbb{K}[[X]]⟶1+X\mathbb{K}[[X]]$ 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 ${\exp }_{!}$ (and its inverse map ${\log }_{!}$) 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 $p$.

25 pages