Certain sets over function fields are polynomial families

TL;DR

This paper proves that special linear groups over polynomial rings in finite fields are polynomial families, extending known results from integers to function fields, with implications for algebraic group theory.

Contribution

It establishes that $ ext{SL}_2$ over polynomial rings in finite fields is a polynomial family, providing a function field analogue of Vaserstein's theorem.

Findings

$ ext{SL}_2( extbf{A})$ is a polynomial family in 52 variables

$ ext{SL}_n( extbf{A})$ is a polynomial family for all $n extgreater 2$

Extension of classical results from integers to function fields

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 $n \ge 3$.