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.