On Separable $\A^2$ and $\A^3$-forms

TL;DR

This paper proves that certain algebraic forms over fields and domains are trivial under specific conditions, extending known results from fields to more general rings.

Contribution

It establishes the triviality of $ ext{A}^3$-forms with locally nilpotent derivations over fields and extends the triviality of separable $ ext{A}^2$-forms to one-dimensional Noetherian domains.

Findings

Any $ ext{A}^3$-form with a suitable derivation over characteristic zero fields is trivial.

Separable $ ext{A}^2$-forms over fields are trivial, extending to one-dimensional Noetherian domains.

The results generalize previous theorems to broader algebraic structures.

Abstract

In this paper, we will prove that any $\A^3$-form over a field $k$ of characteristic zero is trivial provided it has a locally nilpotent derivation satisfying certain properties. We will also show that the result of T. Kambayashi on the triviality of separable $\A^2$-forms over a field $k$ extends to $\A^2$-forms over any one-dimensional Noetherian domain containing $\bQ$.