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Published online by Cambridge University Press: 26 December 2018
In this paper, we will prove that any $\mathbb{A}^{3}$-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 Kambayashi on the triviality of separable
$\mathbb{A}^{2}$-forms over a field
$k$ extends to
$\mathbb{A}^{2}$-forms over any one-dimensional Noetherian domain containing
$\mathbb{Q}$.