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) is an element of 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 circle F-1 = F-1 o 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. Crown Copyright (C) 2022 Published by Elsevier Inc.