Decidability of positive characteristic tame Hahn fields in $\mathcal{L}_t$

We show that any positive characteristic tame Hahn field $\mathbb{F}((t^\Gamma))$ containing $t$ is decidable in $\mathcal{L}_t$, the language of valued fields with a constant symbol for $t$, if $\mathbb{F}$ and $\Gamma$ are decidable. In particular, we obtain decidability of $\mathbb{F}_p((t^{1/p^{\infty}}))$ and $\mathbb{F}_p((t^{\mathbb{Q}}))$ in $\mathcal{L}_t$. This uses a new AKE-principle for equal characteristic tame fields in $\mathcal{L}_t$, building on work by Kuhlmann, together with Kedlaya's work on finite automata and algebraic extensions of function fields. In the process, we obtain an AKE-principle for tame fields in mixed characteristic.