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Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$.

Let $k$ be an algebraically closed field of positive characteristic. Let $A$ be a factorially closed, finitely generated, $k$-sub-algebra of $k[X_1,X_2,X_3]$ such that $tr.deg_k A=2$. Then is it true that $A$ is a polynomial ring in $2$-variables over $k$ i.e. $A$ is isomorphic as a $k$-algebra to $k[T_1,T_2]$ ?

When $k$ is algebraically closed and of characteristic zero, then the answer is affirmative [Theorem 1, Factorially closed subrings of commutative rings, pp.1140]

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  • $\begingroup$ It seems that the same argument works over any positive characteristic. The proof referenced here is using Miyanishi's result, which holds for any characteristic other than 2,3,5. The difference in small characteristic is due to the appearance of more singular surfaces, corresponding to the small subgroups of GL2. When $p=5$ the possibility of $x^2+y^3+z^5$ vanishes, when $p=3$ the binary tetrahedral group ($x^2+y^3+z^4$) appears, and when $p=2$, the binary octahedral group and the binary dihedral groups ($x^2 + y^3 + z^3y$ and $x^2 + y^2z + z^{n-1}$) appear. $\endgroup$
    – assaferan
    Commented May 17, 2018 at 13:43
  • $\begingroup$ They can be eliminated just as in the original argument. $\endgroup$
    – assaferan
    Commented May 17, 2018 at 13:43

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