On Bayart's unique factorization problem

Abstract

Let $A$ be a ring such that the formal power series ring $A[[t]]$ is a unique factorization domain. Bayart's problem asks whether the formal power series ring in two variables $A[[x,t]]$ must also be a UFD. Equivalently, is every irreducible element $f\in A[[x,t]]$ prime? We prove that the answer is yes whenever the constant term of $f$ is an associate of $p^m$, for any prime $p \in A$ and $1\leq m\leq3$, and for $m=4$ if $A$ contains $\Q$. We also prove the following reduction: To prove a positive answer to Bayart's problem, it suffices to prove that $A[x][[t]]$ is a UFD whenever $A[[t]]$ is.