On the degree of the inverse of an automorphism of the affine space

Abstract

We study the degree of the inverse of an automorphism f of the affine n-space over a C -algebra k, in terms of the degree d of f and of other data. For n = 1, we obtain a sharp upper bound for deg ( f − 1) in terms of d and of the nilpotency index of the ideal generated by the coefficients of f ′'. For n = 2 and arbitrary d≥ 3, we construct a (triangular) automorphism f of Jacobian one such that deg( f − 1) > d. This answers a question of A. van den Essen (see [3]) and enables us to deduce that some schemes introduced by authors to study the Jacobian conjecture are not reduced. Still for n = 2, we give an upper bound for deg ( f − 1) when f is triangular. Finally, in the case d = 2 and any n, we complete a result of G. Meisters and C. Olech and use it to give the sharp bound for the degree of the inverse of a quadratic automorphism, with Jacobian one, of the affine 3-space.