On automorphisms of affine superspaces

Abstract

In this note, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field $\mathbb{F}$ of characteristic $0$, which predicts that for a homomorphism $\varphi$ of the polynomial superalgebra $\mathcal{R}:=\mathbb{F}[x_1,\ldots,x_m; \xi_1,\ldots,\xi_m]$ over $\mathbb{F}$, if $\varphi$ satisfies the super version of Jacobian condition (SJ for short), then $\varphi$ gives rise to an automorphism of the affine superspace $\mathbb{A}_{\mathbb{F}}^{m|n}$. We verify the conjecture if additionally, the set $\mathscr{M}$ of maximal $\mathbb{Z}_2$-homogeneous ideals of $\mathcal{R}$ is assumed to be preserved under $\varphi$. The statement is actually proved in any characteristic, i.e. a homomorphism $\varphi$ gives rise to an automorphism of $\mathbb{A}_{\mathbb{F}}^{m|n}$ if SJ is satisfied with $\varphi$ and the set $\mathscr{M}$ is preserved under $\varphi$ for an algebraically closed field $\mathbb{F}$ of any characteristic.