Automorphisms of a polynomial ring which admit reductions of type I

TL;DR

This paper introduces a new method for constructing automorphisms of polynomial rings that admit reductions of type I, expanding understanding of their structure and properties.

Contribution

It presents a novel construction of automorphisms with reductions of type I using locally nilpotent derivations, complementing previous computer-based examples.

Findings

Existence of automorphisms with type I reduction satisfying various degree conditions

New construction method using locally nilpotent derivations

Extension of known examples beyond computationally generated ones

Abstract

Recently, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I--IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen--Makar-Limanov--Willems gave a family of such automorphisms. In this paper, we present a new construction of such automorphisms using locally nilpotent derivations. As a consequence, we discover that there exists an automorphism admitting a reduction of type I which satisfies some degree condition for each possible value.