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 $\text{ML}$ and ${\text{ML}}^{\ast }$ invariants and use them to prove a special case of the strong cancellation of $k^{[2]}$.