A degree bound for strongly nilpotent polynomial automorphisms

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