An Algebraic characterization of the affine three space in arbitrary characteristic

Abstract

We give an algebraic characterization of the affine $3$-space over an algebraically closed field of arbitrary characteristic. We use this characterization to reformulate the following question. Let $$A=k[X,Y,Z,T]/(XY+Z^{p^e}+T+T^{sp})$$ where $p^e\nmid sp$, $sp\nmid p^e$, $e,s\ge 1$ and $k$ is an algebraically closed field of positive characteristic $p$. Is $A=k^{[3]}$? We prove some results on ML and ML${}^{\ast }$ invariants and use them to prove a special case of the strong cancellation of $k^{[2]}$.

arXiv comment12 pages. Changes: A condition missing in the hypothesis of 5.6 was added. Some typos were fixed. To appear in Journal of Commutative Algebra