Rigidity and triangularity of an exponential map

Abstract

Let $k$ be a field of arbitrary characteristic, $A$ be a domain and $K=\mathrm{frac}(A)$. Then (1) All exponential maps of $k^{[3]}$ are rigid, and we give a necessary and sufficient condition for the triangularity of $\delta \in \mathrm{EXP}(k^{[3]})$. (2) If $\delta \in \mathrm{EXP}(A^{[3]})$ such that $\mathrm{rank}(\delta)=\mathrm{rank}(\delta_K)$, then $\delta$ is rigid and we give a necessary and sufficient condition for the triangularity of $\delta$. When $k$ is of zero characteristic, $(1)$ is due to \cite{DD} and $(2)$ is due to \cite{KL}.