Elementary bounded generation for ${\rm SL}_n$ for global function fields and $n\geq 3$

TL;DR

This paper proves that for global function fields, the special linear group SL_n(R) with n≥3 is boundedly generated by elementary matrices, with bounds depending only on n, not on the specific field or ring.

Contribution

It establishes the elementary bounded generation of SL_n(R) over global function fields with bounds independent of the field, extending previous results for number fields.

Findings

Bounded elementary generation for SL_n(R) with n≥3.

Bounds depend only on n, not on the global function field.

Unified bounds for all global function fields.

Abstract

This paper shows that the group ${\rm SL}_n(R)$ is boundedly elementary generated for $n\geq 3$ and $R$ the ring of algebraic integers in a global function field. Contrary to previous statements for number fields and earlier statements for global function fields, the bounds proven in this preprint for elementary bounded generation are independent of the underlying global function field and only depend on the integer $n.$ Combining our main result with earlier results, we further establish that elementary bounded generation always has bounds independent from the global field in question, only depending on $n.$