The freeness property for locally nilpotent derivations of k[x,y,z]

TL;DR

This paper proves Freudenburg's Freeness Conjecture, demonstrating that for any nonzero locally nilpotent derivation of a polynomial ring in three variables over a characteristic zero field, the ring is a free module over the kernel with a basis indexed by natural numbers.

Contribution

It establishes the long-standing Freeness Conjecture for locally nilpotent derivations in three-variable polynomial rings, confirming the structure of the ring as a free module over the kernel.

Findings

B is a free A-module under the derivation D

Existence of a basis with degrees matching natural numbers

Verification of Freudenburg's conjecture

Abstract

We prove Freudenburg's Freeness Conjecture: Let B be the polynomial ring in three variables over a field of characteristic zero, let D : B --> B be a nonzero locally nilpotent derivation, and let A = ker(D). Then B is a free A-module, and there exists a basis $(e_i)_{i \in \mathbb{N}}$ of B such that deg$_D(e_i) = i$ for all $i \in \mathbb{N}$.