A degree bound for strongly nilpotent polynomial automorphisms

Abstract

Let k be a field of characteristic zero. Let F=X+H be a polynomial map from kn to kn, 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)∈kn we haveJH(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 satisfiesdeg⁡(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.