Unimodular rows over affine algebras over algebraic closure of a finite field

TL;DR

This paper proves that unimodular rows over certain affine algebras over algebraic closures of finite fields can be transformed into factorial rows using elementary transformations, under specific conditions.

Contribution

It establishes a new result on the reducibility of unimodular rows over affine algebras over algebraic closures of finite fields.

Findings

Unimodular rows of length equal to the dimension can be transformed into factorial rows.

The result holds for affine algebras of dimension at least 4 over the algebraic closure of finite fields.

The condition 1/(d-1)! in R is crucial for the transformation.

Abstract

In this article, we prove that if $R$ is an affine algebra of dimension $d\geq 4$ over $\overline{\mathbb{F}}_{p}$ and $1/(d-1)! \in R,$ then any unimodular row over $R$ of length $d$ can be mapped to a factorial row by elementary transformations.