Minimal polynomial of an exponential automorphism of C^n

TL;DR

This paper characterizes the minimal polynomial of polynomial exponential automorphisms of complex n-space, showing it has a specific form related to the nilpotency degree of the derivation involved.

Contribution

It provides a precise description of the minimal polynomial for exponential automorphisms in terms of the nilpotency degree of the derivation D.

Findings

The minimal polynomial is of the form (T-1)^d.

d is the smallest integer with D^d(X_i)=0 for all i.

This links the algebraic properties of automorphisms to derivation nilpotency.

Abstract

We show that the minimal polynomial of a polynomial exponential automorphism F of C^n (i.e. F=exp(D), where D is a locally nilpotent derivation) is of the form \mu_F(T)=(T-1)^d, d=min{m \in N: D^m(X_i)=0 for i=1,...,n}.