On exponentials of exponential generating series (Q625352)

The shuffle product of two formal power series \(\sum_{n=0}^{\infty}{\alpha_nX^n}\) and \(\sum_{n=0}^{\infty}{\beta_n X^n}\) with coefficients in a field \(\mathbb K\) is defined by \[ \sum_{n=0}^{\infty}{\gamma_nX^n}:=\sum_{n=0}^{\infty}{\alpha_nX^n} \operatorname{\text Ш} \sum_{n=0}^{\infty}{\beta_nX^n} \] where \[ \gamma_n=\sum_{n=0}^{\infty}{\binom{n}{k}\alpha_k \beta_{n-k}}. \] A result which goes back to Hurwitz shows that the additive group \((X\mathbb K [[X]],+)\) and the shuffle group \((1+X\mathbb K [[X]], \operatorname{\text Ш} )\) are isomorphe. The map which gives this isomorphism is the exponential map defined as follows. If \(A=\sum_{n=1}^{\infty}{\alpha_n X^n}\) and \(B=\sum_{n=1}^{\infty}{\beta_n X^n}\), then \[ \exp_{!}(A):=1+B \] when \[ \exp(\sum_{n=1}^{\infty}{\frac{\alpha_n}{n!} X^n})=1+\sum_{n=1}^{\infty}{\frac{\beta_n}{n!} X^n}. \] In this article, the author proves that the map \(\exp_{!}\) also induces a group isomorphism between the subgroup of rational (respectively algebraic) series of \((X\mathbb K [[X]], +)\) and the subgroup of rational (respectively algebraic) series of \((1+X\mathbb K [[X]], \operatorname{\text Ш} )\), if \(\mathbb K\) is a subfield of the algebraically closed field \(\overline{\mathbb F}_p\) of positive characteristic \(p\). The author also shows that this result is not true if the field is of characteristic zero.