On automorphisms of affine superspaces

TL;DR

This paper explores a super version of the Jacobian conjecture for affine superspaces, proving that certain homomorphisms satisfying a super Jacobian condition and preserving maximal ideals are automorphisms.

Contribution

It introduces a super Jacobian conjecture for affine superspaces and proves it under conditions involving preservation of maximal ideals, extending results to any characteristic.

Findings

Verified the super Jacobian conjecture when maximal ideal set is preserved.

Extended the conjecture's validity to fields of any characteristic.

Established conditions under which homomorphisms are automorphisms.

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.