Certain sets over function fields are polynomial families

Abstract

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 $n\ge 3$.

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