Examples on a Conjecture about Makar-Limanov Invariants of Affine Unique Factorization Domains

TL;DR

This paper explores a conjecture regarding Makar-Limanov invariants in affine UFDs over fields of characteristic zero, demonstrating its limitations over non-algebraically closed fields and providing specific examples.

Contribution

It introduces a conjecture about Makar-Limanov invariants and shows its validity or failure depending on the algebraic closure of the base field, with illustrative examples.

Findings

The conjecture does not always hold over non-algebraically closed fields.

Examples where the conjecture is valid are provided.

The behavior of Makar-Limanov invariants varies with the field's algebraic closure.

Abstract

The author introduces a conjecture about Makar-Limanov invariants of affine unique factorization domains over a field of characteristic zero. Then the author finds that the conjecture does not always hold when $\mathbbm{k}$ is not algebraically closed and gives some examples where the conjecture holds.