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PreprintCC BY 4.0arXiv

On Bayart's unique factorization problem

  • Elad Paran
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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∈ A[[x,t]] prime? We prove that the answer is yes whenever the constant term of f is an associate of pm, for any prime p ∈ A and 1≤ m≤3, 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.

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Cite

BibTeXRIS
BibTeX
@misc{paran2026bayarts,
  title = {{On Bayart's unique factorization problem}},
  author = {Elad Paran},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01622},
  eprint = {2611.01622},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01622},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Bayart's unique factorization problem
AU  - Elad Paran
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01622
UR  - https://arxiv.org/abs/2611.01622
AB  - 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∈ 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 ∈ A and 1≤ m≤3, 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.
ER  -