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In this paper we show that any $\mathbb{A}^2$-fibration over a discretevaluation ring which is also an $\mathbb{A}^2$-form is necessarily a polynomialring. Further we show that separable $\mathbb{A}^2$-forms over PIDs aretrivial.

On the triviality of an $\mathbb{A}^2$-fibration over a DVR