It is a well-known fact that the Henselization of the function field $\mathbb{F}\_{p}(t)$ in regard to the $t$-adic valuation is $\mathbb{F}\_{p}(t)^{alg} \cap \mathbb{F}\_{p}((t))$, so of course $\mathbb{F}\_{p}(t)^{h}$ embeds into $\mathbb{F}\_{p}((t))$, but is it known whether this embedding is elementary in the language of rings $\mathcal{L}\_{\text{Ring}}$ or respectively the language of valued fields $\mathcal{L}\_{\text{Ring}, \ \mid}$ (this does not matter as the valuation is uniformly definable in henselian valued fields with finite residue field). I would assume that, if this is true, it is not known because we do not know whether $\mathbb{F}\_{p}((t))$ is decidable, but do we at least know whether $\mathbb{F}\_{p}(t)^{h}$ is existentially closed in $\mathbb{F}\_{p}((t))$? This at least should be guaranteed for the perfect hull by Ax-Kochen-Ershov for tame fields.



Is $\mathbb{F}_{p}(t)^{h}$ an elementary substructure of/existentially closed in $\mathbb{F}_{p}((t))$?
There is apparently not a very short answer to this. It is, I think, really not known whether this extension is elementary and it would be very surprising if it was known, since we do not even know if the theory of $\mathbb{F}\_{p}((t))$ is decidable. But there is the following result by Franz-Viktor Kuhlmann in his paper "The Algebra and Model Theory of Tame Valued Fields" (<https://arxiv.org/pdf/1304.0194.pdf>, Theorem 5.9) 



> 
> Let $(K,v)$ be a henselian field. Assume that $(L/K,v)$ is a separable subextension of $(K^{c}/K,v)$. Then $(K,v)$ is existentially closed in $(L,v)$. In particular, every henselian inseparable defectless field is existentially closed in its completion.
> 
> 
> 


Here $K^{c}$ stands for the completion in regard to $v$. Now we take a look at $\mathbb{F}\_{p}(t)^{h}$, it is obviously henselian. And from that it obviously follows that any henselian valued field is existentially closed in its completion. Of course this does not give a lot of insight on the problem, so let me try and elaborate a little: The proof of Theorem 5.9 makes use of the fact that it is enough to be existentially closed in every finitely generated subfield to be existentially closed. So we take such a subfield $(F,v)$ and go over to its Henselization, which still lies in the completion (because the completion is Henselian). This is of course a subextension hence it is separable. We can see that indeed $(F,v)^{h}=(K(x\_{1}, \ldots, x\_{n}),v)^{h}$. Then it boils down to show that if we set $K\_{i}:=(K(x\_{1}, \ldots, x\_{i}), v)^{h}$, then $K\_{i} \leq\_{1} K\_{i+1}$, which we can show by realizing that $x$ is the limit of a Cauchy sequence and further results in the linked paper. 


This is of course a very rough sketch. It is better to read the paper to grasp the proof in detail.


