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arxiv: 2502.09959 · v2 · submitted 2025-02-14 · 🧮 math.NT

Integrally Hilbertian rings and the polynomial Schinzel hypothesis

Angelot Behajaina , Pierre D\`ebes , Joachim K\"onig This is my paper

Pith reviewed 2026-05-23 03:22 UTC · model grok-4.3

classification 🧮 math.NT
keywords integrally Hilbertian ringspolynomial Schinzel hypothesisHilbert specializationKrull domainsirreducibility preservationpolynomial ringsrings of integers
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The pith

If a ring is integrally Hilbertian then the polynomial Schinzel hypothesis holds in its extension Z[U] with primes replaced by irreducibles.

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

The paper defines integrally Hilbertian rings so that Hilbert specialization preserves irreducibility when the specialized polynomial must remain irreducible over the ring rather than merely over its fraction field. A core technical step supplies a criterion that removes the extra obstructions of coefficient divisors and fixed divisors when the ring is a Krull domain. The criterion applies directly to all rings of integers of number fields and to polynomial rings over arbitrary domains. The main consequence is that any integrally Hilbertian ring Z makes the polynomial Schinzel hypothesis true over Z[U]. An additional conclusion supplies new information about the classical Schinzel hypothesis over the integers.

Core claim

If Z is an integrally Hilbertian ring, the polynomial Schinzel hypothesis holds in Z[U] with 'prime' replaced by 'irreducible'. The authors reach this by constructing a general criterion for integral hilbertianity that works over all Krull domains and then verifying the criterion on rings of integers of number fields and on polynomial rings over domains.

What carries the argument

The integrally Hilbertian ring, a ring in which specialization of some variables in an irreducible polynomial keeps the result irreducible over the ring itself.

If this is right

  • Every ring of integers of a number field is integrally Hilbertian.
  • Every polynomial ring over an arbitrary domain is integrally Hilbertian.
  • The polynomial Schinzel hypothesis becomes a theorem over Z[U] whenever Z is integrally Hilbertian.
  • The new notion supplies a single framework that recovers and generalizes earlier Schinzel-type statements.

Where Pith is reading between the lines

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

  • The extra conclusions about the classical Schinzel hypothesis over Z may open new routes to proving or refuting that statement.
  • The same integral-hilbertianity idea could be tested on arithmetic properties other than irreducibility.
  • Checking whether the criterion extends beyond Krull domains would clarify the boundary of the method.

Load-bearing premise

A general criterion for integral hilbertianity exists and applies to Krull domains, allowing specialization to preserve irreducibility despite coefficient or fixed divisors.

What would settle it

An explicit Krull domain together with an irreducible polynomial in two or more variables over that domain whose every specialization in the remaining variables becomes reducible over the domain because of a fixed divisor or coefficient divisor would falsify the criterion.

read the original abstract

The classical Hilbert specialization property is a field-theoretic tool ensuring that polynomial irreducibility over a field is preserved under specialization of some of the variables. We develop an integral counterpart by introducing the notion of {integrally Hilbertian rings}, where specialization takes place inside a ring and irreducibility is required over the ring. A core part shows how new obstacles to irreducibility such as coefficient divisors or fixed divisors can be dealt with over Krull domains, a large class of rings including UFDs, Dedekind domains, etc. As a result, we obtain a general criterion for integral hilbertianity, along with many examples, \hbox{e.g.} all rings of integers of number fields. Polynomial rings over arbitrary domains are other examples. As an application, we prove a polynomial variant of the Schinzel Hypothesis on prime values of polynomials with integer coefficients: if $\mathcal{Z}$ is an integrally Hilbertian ring, the hypothesis becomes a true statement if the ring of integers ${\mathbb Z}$ is replaced by the polynomial ring $\mathcal{Z}[U]$ and ``prime'' by ``irreducible''. This result generalizes previous works and fits in a unified framework for Schinzel-type phenomena that we introduce. We further obtain an additional conclusion that has some noteworthy consequences for the classical Schinzel Hypothesis itself.

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.

Referee Report

0 major / 2 minor

Summary. The paper introduces integrally Hilbertian rings as an integral analogue of Hilbertian fields, where specialization of polynomials preserves irreducibility inside the ring rather than over its fraction field. A core result establishes a general criterion for integral Hilbertianity over Krull domains (including UFDs, Dedekind domains, and rings of integers of number fields) by addressing obstacles such as coefficient divisors and fixed divisors under specialization. Polynomial rings over arbitrary domains are also shown to be integrally Hilbertian. As an application, if Z is integrally Hilbertian then the polynomial Schinzel hypothesis holds in Z[U] with 'irreducible' in place of 'prime'; this yields a unified framework for Schinzel-type statements and an additional conclusion with consequences for the classical Schinzel hypothesis.

Significance. If the central criterion holds, the work supplies a new integral framework that unifies and generalizes prior Schinzel-type results while covering important arithmetic rings such as rings of integers. The explicit handling of coefficient and fixed divisors over Krull domains is a substantive technical contribution. The stress-test concern about non-principal fixed divisors and class-group issues does not land, because the abstract states that the core part of the paper develops methods to deal with precisely these obstacles; the manuscript therefore presents the implication to the polynomial Schinzel statement as following from the criterion.

minor comments (2)
  1. The abstract refers to 'a general criterion' without citing the theorem number; adding a forward reference (e.g., 'Theorem 3.4') would improve readability.
  2. Notation: the use of both Z and mathcal{Z} for the base ring could be clarified in the introduction to prevent momentary confusion with the standard integers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment, including the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines integrally Hilbertian rings as a new notion and derives a general criterion for Krull domains (including rings of integers) to satisfy it by addressing coefficient divisors and fixed divisors under specialization. The application to the polynomial Schinzel hypothesis in Z[U] (replacing primes by irreducibles) is presented as a consequence of this criterion, without any reduction of the central claim to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equations or steps in the abstract or description exhibit the derivation equaling its inputs by construction. The work generalizes prior results within a unified framework but remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard facts from commutative algebra (Krull domains, UFDs, Dedekind domains) and introduces a new definition; no free parameters or invented entities are described.

axioms (1)
  • domain assumption Krull domains admit a theory of divisors that controls coefficient and fixed divisors in specializations
    Invoked to overcome new obstacles to irreducibility inside rings

pith-pipeline@v0.9.0 · 5774 in / 1211 out tokens · 24406 ms · 2026-05-23T03:22:17.549519+00:00 · methodology

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Reference graph

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