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On Landweber`s unique factorization problem

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Pith's one-line read Polynomial rings in any number of variables over a regular UFD remain unique-factorization domains after adjoining formal power series in one variable.

desk verdict Solid solution to Landweber’s 1974 question: finite-height irreducibility for Krull domains plus a clean retraction criterion yields R[[t]] UFD for infinite-variable polynomials over regular UFDs. read the letter →

arxiv 2607.03475 v2 pith:R6ICMMEQ submitted 2026-07-03 math.AC

Adam Jones , Elad Paran This is my paper · ORCID
classification math.AC MSC 13F2513F1513A05
keywords uniquefactorizationdomainformalpowerseriesKrullfiniteirreducibilityheightregularUFDLandweberproblemC-primeelements
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

For fifty years it has been open whether the formal power series ring over a polynomial ring in infinitely many variables is still a unique-factorization domain. The paper answers yes: if A is a regular UFD and R is the polynomial ring over A in any set of variables, then R[[t]] is a UFD. The argument rests on a general finite-height theorem: every irreducible power series over a Krull domain is already irreducible modulo some finite power of t. Once that is known, a finite-stage retraction criterion reduces the infinite-variable case to the classical Samuel–Buchsbaum theorem for regular UFDs in finitely many variables. The result settles Landweber’s 1974 question and supplies a uniform reason why unique factorization survives the passage to power series for this large class of rings.

What carries the argument

The finite irreducibility theorem (Theorem B): an irreducible f in R[[t]] for R Krull is already irreducible modulo t^n for some n. It is proved by establishing quantitative C-primality over DVRs, then extracting a global factorization via a König-lemma argument on finitely many candidate ideals.

What would settle it

Exhibit a single irreducible power series over a discrete valuation ring that factors non-trivially modulo every power of t, or exhibit a regular UFD A and an infinite set of variables such that some irreducible of finite height in A[x_i][[t]] fails to be prime.

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Extended reading notes

Core claim

If R is any Krull domain then every irreducible element of R[[t]] is irreducible modulo some finite power of t. Combined with a retraction criterion that reduces finite sets of coefficients to a UFD subring, this implies that the power series ring over a polynomial ring in arbitrarily many variables over a regular UFD is itself a UFD.

Load-bearing premise

The recursive bounds that guarantee only finitely many candidate factor ideals appear at each valuation level over a discrete valuation ring must hold; if those bounds fail, the extraction of a global factorization collapses.

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Desk editor's note, referee report, and a circularity audit.

Referee Report

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Summary. The paper solves Landweber’s 1974 question by proving that if A is a regular UFD and R = A[x_i | i ∈ I] for an arbitrary index set I, then R[[t]] is a UFD (Theorem A / 7.3). The argument rests on two general results for Krull domains: (B) every irreducible f ∈ R[[t]] is irreducible modulo some finite power of t (Theorem 6.4), and (C) a finite-stage retraction criterion that reduces the UFD property of R[[t]] to the classical Samuel–Buchsbaum theorem for regular UFDs (Theorem 7.2). The bulk of the work is the proof of (B): first a quantitative C-primality theory over DVRs (§§4–5) that produces only finitely many candidate ideals of the form (A, t^ℓ), then a König-lemma extraction of a global factorization (Lemma 3.1), and finally a local-to-global comparison of those ideals via height-1 localizations (Proposition 6.3 + Lemma 6.2).

Significance. Landweber’s problem has been open for fifty years and is repeatedly cited as a basic open question about unique factorization in non-Noetherian power series rings. The paper settles it completely, and the intermediate finite-height theorem (Theorem B) is of independent interest for arbitrary Krull domains. The reduction via retractions cleanly isolates the new work from the classical Samuel–Buchsbaum theorem, and the C-primality machinery supplies an explicit, elementary substitute for Artin approximation in the DVR case. These are substantial, self-contained advances in commutative algebra.

minor comments (5)
  1. The recursive definition of the control constants a^{(m)}_k(L) and N_m (after Lemma 4.8 and before Proposition 4.12) is correct but dense; a short remark that the only inputs are the valuation inequalities of Lemma 4.5 and the length bound of Lemma 4.10 would help the reader track the induction.
  2. In the remark at the end of §4 the counter-example a = p^{2} - Y t^{2} over Q[X,Y] is asserted without verification. A one-line check that it is irreducible mod t^{3} yet fails C-primality for every C would make the sharpness claim fully self-contained.
  3. Lemma 6.1 (finitely many divisors up to associates in a Krull domain) is standard; a parenthetical reference to Bourbaki VII.1 would be useful for non-specialists.
  4. Typographical: “K¨onig’s lemma” appears with inconsistent diacritics; standardize to “König’s lemma” throughout.
  5. The final open question (if R[[x]] is a UFD, is R[[x]][[t]] a UFD?) is well-posed; a brief pointer to Bayart’s earlier formulation would complete the historical picture.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: finite-height irreducibility and the UFD conclusion are derived from DVR valuation bounds, König extraction, and the external Samuel–Buchsbaum theorem.

full rationale

The derivation is self-contained. Theorem B is proved by contraposition: assume factorizations modulo arbitrarily high powers of t; the quantitative C-primality constants N_m (built inductively from the elementary valuation inequalities of Lemma 4.5 and the length bound k_d(W)<v(W_0) of Lemma 4.10) guarantee only finitely many candidate ideals (A,t^ℓ) for large n (Lemma 5.1); König’s lemma (Lemma 3.1) then extracts a global factorization, a contradiction. The local-to-global step (Proposition 6.3 + Lemma 6.2) uses only the definition of a Krull domain. Theorem C reduces the infinite-variable case to finite polynomial rings via retractions; those rings are regular UFDs, so their power-series rings are UFDs by the classical Samuel–Buchsbaum theorem, cited as external literature. No parameter is fitted to data, no uniqueness theorem is imported from the authors’ prior work, and no equation is forced by a normalization chosen to produce the result. The only external inputs are standard facts about Krull domains and the Samuel–Buchsbaum theorem.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper works entirely inside standard commutative algebra. No free parameters are fitted. The only external non-standard inputs are the classical Samuel–Buchsbaum theorem (already published) and the definition of Krull domain. All other ingredients (C-primality, control constants, König extraction, retractions) are constructed inside the paper.

assumptions (3)
  • domain assumption Samuel–Buchsbaum theorem: if R is a regular UFD then R[[t]] is a UFD (used for finite-variable subrings).
    Invoked in the proof of Theorem 7.3 and in the introduction; treated as a black-box external result.
  • standard math Definition and basic properties of Krull domains (intersection of DVRs, finite support of valuations, atomicity of R[[t]]).
    Recalled in §6 from Bourbaki; used throughout the globalization argument.
  • standard math König's lemma for infinite finitely branching trees.
    Applied in Lemma 3.1 to extract a coherent factorization from compatible partial factorizations.
invented entities (1)
  • C-prime elements (and the associated control constants N_m, a^{(m)}_k(L))
    purpose: Quantitative finite-height primality that lets the authors bound the number of candidate ideals (A,t^ℓ) and feed the König extraction.
    Defined in §4; the recursive bounds are constructed ad hoc for the proof and have no independent external existence.

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Pith. "Pith review of On Landweber`s unique factorization problem." pith.science (2026). https://pith.science/paper/R6ICMMEQ

@misc{pith2026260703475,
  author       = {Pith},
  title        = {Pith review of: On Landweber`s unique factorization problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R6ICMMEQ}},
  note         = {Machine review of arXiv:2607.03475}
}
abstract

We solve a long-standing open problem, posed by Landweber in 1974: Let $R = K[x_1, x_2, . . . ]$ be the ring of polynomials in countably many variables over a field $K$. Is the formal power series ring $R[[t]]$ a unique factorization domain? We prove that it is. The proof is based on a new general result in commutative algebra: If $R$ is a Krull domain, and $f \in R[[t]]$ is irreducible, then $f$ is irreducible modulo a finite power of $t$.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The Noetherian Case of Bayart's Power-Series Question

    math.AC 2026-08 conditional novelty 7.0 of 10

    For a commutative Noetherian ring R, if R[[x]] is a unique factorization domain, then R[[x,y]] is also a unique factorization domain.

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