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 of characteristic 0, which predicts that for a homomorphism phi of the polynomial superalgebra R := [x(1), .... ,x(m); xi(1), ... , xi(m)] over , if phi satisfies the super version of Jacobian condition (SJ for short), then phi gives rise to an automorphism of the affine superspace (m|n). We verify the conjecture if additionally, the set of maximal Z(2)-homogeneous ideals of R is assumed to be preserved under phi. The statement is actually proved in any characteristic, i.e. a homomorphism phi gives rise to an automorphism of (m|n) if SJ is satisfied with phi and the set is preserved under phi for an algebraically closed field of any characteristic.