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 $\ddot{I}†$ of the polynomial superalgebra $\mathcal{R}:=\mathbb{F}[x_1,\dots ,x_m;\hat{I}{¾}_1,\dots ,\hat{I}{¾}_m]$ over $\mathbb{F}$, if $\ddot{I}†$ satisfies the super version of Jacobian condition (SJ for short), then $\ddot{I}†$ gives rise to an automorphism of the affine superspace ${\mathbb{A}}_{\mathbb{F}}^{m|n}$. We verify the conjecture if additionally, the set $\mathcal{M}$ of maximal ${\mathbb{Z}}_2$-homogeneous ideals of $\mathcal{R}$ is assumed to be preserved under $\ddot{I}†$. The statement is actually proved in any characteristic, i.e. a homomorphism $\ddot{I}†$ gives rise to an automorphism of ${\mathbb{A}}_{\mathbb{F}}^{m|n}$ if SJ is satisfied with $\ddot{I}†$ and the set $\mathcal{M}$ is preserved under $\ddot{I}†$ for an algebraically closed field $\mathbb{F}$ of any characteristic.

8 pages. To appear in JAA