A characterization of local nilpotence for dimension two polynomial derivations

TL;DR

This paper characterizes when a polynomial derivation in two variables over an algebraically closed field is locally nilpotent, linking this property to the existence of commuting automorphisms of arbitrarily large degree.

Contribution

It provides a new criterion for local nilpotence of two-variable polynomial derivations based on the structure of commuting automorphisms.

Findings

A polynomial derivation is locally nilpotent iff its commuting automorphism subgroup contains elements of arbitrarily large degree.

Establishes a precise equivalence between local nilpotence and the automorphism subgroup structure.

Advances understanding of polynomial derivations and automorphism groups in two variables.

Abstract

Let K be an algebraically closed field. We prove that a polynomial K-derivation $D$ in two variables is locally nilpotent if and only if the subgroup of polynomial K-automorphisms which commute with D admits elements whose degree is arbitrary big.