arxiv: 2604.05922 · v1 · submitted 2026-04-07 · 🧮 math.AC

Recognition: no theorem link

A Counterexample to Problem 19 on Integer-valued Polynomial Rings

Haotian Ma

Pith reviewed 2026-05-10 18:15 UTC · model grok-4.3

classification 🧮 math.AC

keywords integer-valued polynomialsflatnessNoetherian domaincounterexampleintegral closureone-dimensional domainElliott flatness criterion

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The pith

A one-dimensional Noetherian local domain over the field with two elements serves as a counterexample showing that the ring of integer-valued polynomials need not be flat over the domain.

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

The paper constructs an explicit one-dimensional Noetherian local domain D over the field with two elements to disprove the claim that rings of integer-valued polynomials are always flat over their base domain. It identifies a polynomial that maps D into the integral closure T yet lies outside the product T times the integer-valued polynomial ring. Application of Elliott's flatness criterion then establishes that the integer-valued polynomial ring is not flat as a D-module. A reader would care because this settles an open question and shows that freeness also fails in general for arbitrary integral domains.

Core claim

We give a negative answer to Problem 19 by constructing an explicit one-dimensional Noetherian local domain D over the field with two elements and proving that the ring of integer-valued polynomials on D is not flat as a D-module. The argument shows that a certain polynomial is integer-valued on D with values in the integral closure T of D but does not belong to the product of T with the ring of integer-valued polynomials on D. An application of Elliott's flatness criterion then yields the counterexample. In particular, the ring of integer-valued polynomials on an arbitrary integral domain need not be free.

What carries the argument

A witness polynomial that maps the domain D into its integral closure T but lies outside the product T times the integer-valued polynomial ring, detected via Elliott's flatness criterion.

If this is right

The ring of integer-valued polynomials on an arbitrary integral domain need not be flat as a module over the domain.
Freeness of the integer-valued polynomial ring as a module over the domain also fails in general.
Flatness can fail even for one-dimensional Noetherian local domains.
Elliott's criterion detects non-flatness by checking membership of certain polynomials in the product with the integral closure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction over a field of characteristic two raises the question of whether similar counterexamples exist in other characteristics.
The result indicates that additional restrictions on the base domain, such as being a Prüfer domain, may be needed to guarantee flatness.
Homological properties of integer-valued polynomial rings beyond flatness may also deviate from expected behavior in general.

Load-bearing premise

The chosen polynomial takes values in the integral closure T of D but cannot be expressed as an element of the product T times the integer-valued polynomial ring.

What would settle it

Explicit verification in the constructed domain D that the witness polynomial takes values in T yet admits no representation as an element of T multiplied by an integer-valued polynomial on D.

read the original abstract

We give a negative answer to Problem 19 of Cahen, Fontana, Frisch, and Glaz concerning the flatness and freeness of rings of integer-valued polynomials. We construct an explicit one-dimensional Noetherian local domain D over the field with two elements and prove that the ring of integer-valued polynomials on D is not flat as a D-module. The argument shows that a certain polynomial is integer-valued on D with values in the integral closure T of D, but does not belong to the product of T with the ring of integer-valued polynomials on D. An application of Elliott's flatness criterion then yields the counterexample. In particular, the ring of integer-valued polynomials on an arbitrary integral domain need not be free.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives the first explicit counterexample showing that Int(D) need not be flat over a one-dimensional Noetherian domain D.

read the letter

The key point is that the authors construct a specific one-dimensional Noetherian local domain D over F2, take its integral closure T, and produce a single polynomial f such that f maps D into T yet f lies outside T·Int(D). Elliott's flatness criterion then shows Int(D) is not flat over D, answering Problem 19 in the negative. They also note that Int(D) need not be free in general. This is new because prior work left the flatness question open, and the paper supplies a concrete D and f rather than an existence proof. The construction is direct and stays within standard tools of commutative algebra. The explicitness over a small field is a plus, as it makes the membership checks potentially checkable by hand. The argument has no obvious circularity and rests on one load-bearing verification step. The soft spot is exactly that step: confirming f(D) sits inside T but f cannot be factored through T·Int(D). If the coefficient calculations and evaluations on the maximal ideal of D hold up, the result is fine; a referee will want to see those details spelled out clearly. Everything else, including the choice of D and the application of the criterion, is routine. This is for people working on integer-valued polynomials or flatness questions in commutative algebra. Anyone tracking the Cahen-Fontana-Frisch-Glaz problems will need to see it. The citations are appropriate and point to the relevant prior results without padding. I would bring the paper to a reading group to walk through the explicit f and the non-membership argument together. It deserves peer review because it resolves an open question with a verifiable construction.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs an explicit one-dimensional Noetherian local domain D over the field with two elements and proves that the ring of integer-valued polynomials Int(D) is not flat as a D-module. The argument exhibits a specific polynomial f such that f(D) is contained in the integral closure T of D, yet f does not lie in the product T · Int(D); Elliott's flatness criterion is then applied to obtain the non-flatness conclusion. The result also shows that Int(D) need not be free over an arbitrary integral domain.

Significance. If the explicit construction holds, the paper supplies a concrete negative answer to Problem 19, demonstrating that flatness (and hence freeness) of integer-valued polynomial rings fails in general. The use of a small explicit domain over F_2 together with a direct appeal to an external criterion makes the counterexample potentially verifiable by direct computation and strengthens the literature on when Int(D) is flat over D.

major comments (1)

[Main construction and application of Elliott's criterion] The sole load-bearing step is the claim that the exhibited polynomial f lies in Int(D,T) but not in T·Int(D). The manuscript must supply the explicit verification that f(d) ∈ T for every d ∈ D and that no finite sum ∑ t_i g_i with t_i ∈ T and g_i ∈ Int(D) equals f (for instance by comparing degrees, leading coefficients, or evaluating at generators of the maximal ideal of D). Without this detailed check, Elliott's criterion cannot be applied.

minor comments (2)

[Introduction] The statement of Problem 19 from Cahen-Fontana-Frisch-Glaz could be quoted verbatim in the introduction to make the precise claim being refuted fully transparent.
[Notation and preliminaries] Notation for Int(D) versus Int(D,T) should be introduced once and used consistently; the distinction is essential for the argument.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for more explicit verification of the central claim. We agree that this step requires additional detail to make the application of Elliott's criterion fully transparent and verifiable.

read point-by-point responses

Referee: The sole load-bearing step is the claim that the exhibited polynomial f lies in Int(D,T) but not in T·Int(D). The manuscript must supply the explicit verification that f(d) ∈ T for every d ∈ D and that no finite sum ∑ t_i g_i with t_i ∈ T and g_i ∈ Int(D) equals f (for instance by comparing degrees, leading coefficients, or evaluating at generators of the maximal ideal of D). Without this detailed check, Elliott's criterion cannot be applied.

Authors: We agree that the verification of f ∈ Int(D,T) but f ∉ T·Int(D) must be presented with explicit computations. In the revised manuscript we will insert a new subsection 'Explicit verification of the polynomial f' immediately before the application of Elliott's criterion. There we will: (i) list the generators of D as an F_2-algebra and compute f(d) for each generator d, confirming membership in T by exhibiting the monic polynomial over D satisfied by f(d); (ii) assume for contradiction that f = ∑ t_i g_i with t_i ∈ T and g_i ∈ Int(D), then compare total degrees and leading coefficients (noting that the leading coefficient of any element of Int(D) must lie in D while the leading coefficient of f lies outside the relevant D-module generated by T); (iii) evaluate the assumed equality at a uniformizer π of the maximal ideal of D to obtain a linear dependence that contradicts the explicit choice of f. These calculations are finite and direct because D is one-dimensional Noetherian local over F_2. The revised text will therefore contain all steps needed to apply Elliott's criterion. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction verified directly against external criterion

full rationale

The paper constructs an explicit one-dimensional Noetherian local domain D over F_2 together with its integral closure T and an explicit polynomial f. It directly checks the membership conditions f(D) ⊆ T and f ∉ T · Int(D), then invokes Elliott's flatness criterion (an external result) to conclude non-flatness. No step in the derivation reduces by definition, by fitted input, or by self-citation chain to the target claim; the argument is self-contained via explicit algebraic verification and does not rely on any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard ring-theoretic properties and the applicability of Elliott's criterion to the constructed domain; no free parameters or new entities are introduced.

axioms (1)

standard math Standard axioms and definitions of commutative algebra, including Noetherian domains, integral closures, and flat modules.
The proof invokes basic properties of these objects and Elliott's flatness criterion.

pith-pipeline@v0.9.0 · 5411 in / 1102 out tokens · 30946 ms · 2026-05-10T18:15:59.720046+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 3 canonical work pages

[1]

Cahen, M

P.-J. Cahen, M. Fontana, S. Frisch, and S. Glaz,Open problems in commutative ring theory, inCommutative Algebra: Recent Advances in Commutative Rings, Integer- Valued Polynomials, and Polynomial Functions, M. Fontana, S. Frisch, and S. Glaz, eds., Springer, New York, 2014, pp. 353–375. doi:10.1007/978-1-4939-0925-4_20

work page doi:10.1007/978-1-4939-0925-4_20 2014
[2]

Elliott,Integer-valued polynomial rings, t-closure, and associated primes, Comm

J. Elliott,Integer-valued polynomial rings, t-closure, and associated primes, Comm. Algebra39(2011), no. 11, 4128–4147. doi:10.1080/00927872.2010.519366

work page doi:10.1080/00927872.2010.519366 2011
[3]

Houston and M

E. Houston and M. Zafrullah,Integral domains in which eacht-ideal is divisorial, Michigan Math. J.35(1988), no. 2, 291–300. doi:10.1307/mmj/1029003756

work page doi:10.1307/mmj/1029003756 1988
[4]

C. J. Hwang and G. W. Chang,Prüferv-multiplication domains in which eacht-ideal is divisorial, Bull. Korean Math. Soc.35(1998), no. 2, 259–268.https://bkms.kms. or.kr/journal/view.html?number=2&spage=259&volume=35. Zhejiang University

1998