A degree bound for strongly nilpotent polynomial automorphisms

TL;DR

This paper establishes a new degree bound for the inverse of polynomial automorphisms with strongly nilpotent Jacobian matrices, showing it is bounded by the degree of the original map raised to the power of the nilpotency index.

Contribution

It proves a tighter degree bound for inverses of strongly nilpotent polynomial automorphisms, improving upon the general bound for all polynomial automorphisms.

Findings

Degree of inverse is at most deg(F)^p for strongly nilpotent cases.

Improves the known bound from deg(F)^n to deg(F)^p.

Applicable over fields of characteristic zero.

Abstract

Let $k$ be a field of characteristic zero. Let $F = X + H$ be a polynomial mapping from $k^n \to k^n$, where $X$ is the identity mapping 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 $X^{(1)},\ldots,X^{(p)} \in k^n$ we have \begin{align*} JH(X^{(1)})\ldots JH (X^{(p)}) = 0. \end{align*} Every $F$ of this form is a polynomial automorphism, i.e. there is a second polynomial mapping $F^{-1}$ such that $F \circ F^{-1} = F^{-1} \circ F = X$. We prove that the degree of the inverse $F^{-1}$ satisfies \begin{align*} deg(F^{-1}) \leq deg(F)^p, \end{align*} improving in the strongly nilpotent case on the well known degree bound $deg(F^{-1}) \leq deg(F)^n$ for general polynomial automorphisms.