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

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 $n.$