On automorphisms of affine superspaces

In this paper, we propose a super version of Jacobian conjecture on the automorphisms of affine superspaces over an algebraically closed field F of characteristic 0, which predicts that for a homomorphism φ of the polynomial superalgebra ℛ:= F[x₁,…,x_m; ξ₁,…,ξ_m] over F, if φ satisfies the super version of Jacobian condition (SJ for short), then φ gives rise to an automorphism of the affine superspace A_F^m|n. We verify the conjecture if additionally, the set M of maximal Z₂-homogeneous ideals of R is assumed to be preserved under φ. The statement is actually proved in any characteristic, i.e. a homomorphism φ gives rise to an automorphism of A_F^m|n if SJ is satisfied with φ and the set M is preserved under φ for an algebraically closed field F of any characteristic.