On exponentials of exponential generating series

Details On exponentials of exponential generating series On exponentials of exponential generating series

2008-07-03

arxiv.org/abs/0807.0540v4

Algebra \& Number Theory 4, 7 (2010) 919-942

Mathematics

Number Theory

25 pages

Scientific paper

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 $p$.

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