A degree bound for strongly nilpotent polynomial automorphisms

Let k be a field of characteristic zero. Let F=X+H be a polynomial map from k ⁿ to k ⁿ, where X is the identity map and H has only degree two terms and higher. We say that the Jacobian matrix JH of H is strongly nilpotent with index p if for all X ⁽¹⁾,…,X ^(p)∈k ⁿ we have JH(X ⁽¹⁾)…JH(X ^(p))=0. Every F of this form is a polynomial automorphism, i.e. there is a second polynomial map F ⁻¹ such that F∘F ⁻¹=F ⁻¹∘F=X. We prove that the degree of the inverse F ⁻¹ satisfies deg⁡(F ⁻¹)≤deg⁡(F) ^p−1, improving in the strongly nilpotent case on the well known degree bound deg⁡(F ⁻¹)≤deg⁡(F) ⁿ⁻¹ for general polynomial automorphisms.