Elimination results for tame fields with finite residue fields

Building on work of Kuhlmann and Lisinski, we study the theory of the Hahn series field ${\mathbb{F}}_q(\text{\hspace{0.17em}⁣}(\mathbb{Q})\text{\hspace{0.17em}⁣})$, over a finite field ${\mathbb{F}}_q$, equipped with the $t$-adic valuation, in a language of valued fields. We prove that every formula is equivalent to a formula $\mathrm{\exists }y\text{\hspace{0.17em}⁣}:f(x_1,\dots ,x_n,y)=0$, for a polynomial $f\in \mathbb{Z}[x_1,\dots ,x_n,y]$.