A note on $\mathbb{G}_a$-actions in positive characteristic

Miyanishi proved that the ring of invariants of any $\mathbb{G}_a$ action on $\mathbb{A}^3$ is $\mathbb{A}^2$, when the field $k$ has zero characteristic. However, it is not known if this result holds when $k$ has positive characteristic. We provide a sufficient condition under which this result holds in positive characteristic. We also prove the following results related to the rigidity of the ring of invariants of an exponential map of a polynomial ring. (1) 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. (2) 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. When $k$ is an algebraically closed of zero characteristic, the commuting derivations conjecture for $k^{[3]}$ has been proved by Maubach and El Kahoui proved that the weak Abhyankar Sathaye conjecture is equivalent to the commuting derivations conjecture. By introducing the notion of commuting exponential maps and formulating the commuting exponential maps conjecture, we show that the weak Abhyankar-Sathaye conjecture is equivalent to the commuting exponential maps conjecture for any field of arbitrary characteristic. In particular, we prove the commuting derivations conjecture$(CD(3))$ for any field of zero characteristic.