  In 1938, Skolem conjectured that $\mathbf{SL}_n(\mathbb{Z})$ is not a
polynomial family for any $n \ge 2$. Carter and Keller disproved Skolem's
conjecture for all $n \ge 3$ by proving that $\mathbf{SL}_n(\mathbb{Z})$ is
boundedly generated by the elementary matrices, and hence a polynomial family
for any $n \ge 3$. Only recently, Vaserstein refuted Skolem's conjecture
completely by showing that $\mathbf{SL}_2(\mathbb{Z})$ is a polynomial family.
An immediate consequence of Vaserstein's theorem also implies that
$\mathbf{SL}_n(\mathbb{Z})$ is a polynomial family for any $n \ge 3$. In this
paper, we prove a function field analogue of Vaserstein's theorem: that is, if
$\mathbf{A}$ is the ring of polynomials over a finite field of odd
characteristic, then $\mathbf{SL}_2(\mathbf{A})$ is a polynomial family in 52
variables. A consequence of our main result also implies that
$\mathbf{SL}_n(\mathbf{A})$ is a polynomial family for any $n \ge 3$.
