Commuting Exponential maps
Abstract: Let $k$ be a field of arbitrary characteristic. We introduce the notion of commuting exponential maps and formulate the commuting exponential maps conjecture. We show that the weak Abhyankar-Sathaye conjecture is equivalent to the commuting exponential maps conjecture. In particular, we prove that the commuting derivations conjecture $CD(3)$ is true for any field of zero characteristic. We also prove the following results related to ring of invariants of an exponential map of a polynomial ring. (1) Let $B=k{[3]}$ and $\delta \in \mathrm{EXP}(B)$ be non-trivial. If the plinth ideal $\mathrm{pl} (\delta)$ contains a quasi-basic element, then the ring of invariants ($B{\delta}$) is $k{[2]}$. (2) Let $B=R{[n]}$, where $R$ is a $k$-domain and $\delta \in \mathrm{EXP}_R(B)$ is a triangular exponential map. Then $B{\delta}$ is non-rigid. In particular, for any field $k$ of zero characteristic the kernel of any triangular $R$-derivation of $R{[n]}$ is non-rigid. (3) Let $k$ be a field of zero characteristic and $R$ be a $k$-domain. Then the kernel of any linear locally nilpotent $R$-derivation of $R{[n]}$ is non-rigid.
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