Let $G$ be a linear algebraic group. Let $k$ be a field, let $k(t)$ be the field of fractions of the polynomial algebra $k[t]$ and let $k((t))$ be the field of Laurent series. I'm interested in knowing whether for each element $g\in G(k((t)))$ there is a sequence $(g_n)_n$ of elements $g_n\in G(k(t))$ such that each entry of $g_n$ converges to the corresponding entry of $g$ for the $t$-adic topology.
Going through the literature I found, for general systems of equations, the algebraic Artin approximation theorem. But it seems that this theorem is concerned with local rings $(A,\mathfrak m)$ and their completions with respect to the $\mathfrak m$-adic topology, and it is not clear to me whether this can be used to answer my question.