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 [Formula: see text] of characteristic [Formula: see text], which predicts that for a homomorphism [Formula: see text] of the polynomial superalgebra [Formula: see text] over [Formula: see text], if [Formula: see text] satisfies the super version of Jacobian condition (SJ for short), then [Formula: see text] gives rise to an automorphism of the affine superspace [Formula: see text]. We verify the conjecture if additionally, the set [Formula: see text] of maximal [Formula: see text]-homogeneous ideals of [Formula: see text] is assumed to be preserved under [Formula: see text]. The statement is actually proved in any characteristic, i.e. a homomorphism [Formula: see text] gives rise to an automorphism of [Formula: see text] if SJ is satisfied with [Formula: see text] and the set [Formula: see text] is preserved under [Formula: see text] for an algebraically closed field [Formula: see text] of any characteristic.