A commuting derivations theorem on UFDs

TL;DR

This paper proves a version of the commuting derivations conjecture for polynomial rings over UFDs, showing under certain conditions that the ring is a polynomial ring and the invariant is a coordinate, linking to the Sataye conjecture.

Contribution

It extends the commuting derivations conjecture to UFDs with less restrictive assumptions and proves the polynomiality and coordinate property under additional conditions.

Findings

Proved a version of the commuting derivations conjecture for UFDs.

Showed that under certain conditions, the ring is a polynomial ring and the invariant is a coordinate.

Linked the conjecture to the Sataye conjecture and provided a new equivalent formulation.

Abstract

Let $A$ be the polynomial ring over $k$ (a field of characteristic zero) in $n+1$ variables. The commuting derivations conjecture states that $n$ commuting locally nilpotent derivations on $A$, linearly independent over $A$, must satisfy $A^{D_1,...,D_m}=k[f]$ where $f$ is a coordinate. The conjecture can be formulated as stating that a $(G_m)^n$-action on $k^{n+1}$ must have invariant ring $k[f]$ where $f$ is a coordinate. In this paper we prove a statement (theorem \ref{CDH2}) where we assume less on $A$ ($A$ is a {\sc UFD} over $k$ of transcendence degree $n+1$ satisfying $A^*=k$) and prove less ($A/(f-\alpha)$ is a polynomial ring for all but finitely many $\alpha$). Under certain additional conditions (the $D_i$ are linearly independent modulo $(f-\alpha)$ for each $\alpha\in k$) we prove that $A$ is a polynomial ring itself and $f$ is a coordinate. This statement is proven even more generally by replacing ``free unipotent action of dimension $n$'' for ``$G_a^n$-action''. We make links with the (Abhyankar-)Sataye conjecture and give a new equivalent formulation of the Sataye conjecture.