A generalization of Dumas irreducibility criterion

Jitender Singh

arxiv: 2505.08549 · v3 · pith:PCZRHRFInew · submitted 2025-05-13 · 🧮 math.NT · math.AC· math.AG

A generalization of Dumas irreducibility criterion

Jitender Singh This is my paper

Pith reviewed 2026-05-22 15:49 UTC · model grok-4.3

classification 🧮 math.NT math.ACmath.AG

keywords Newton polygonsDumas criterionirreducibilitydiscrete valuation domainspolynomial factorizationnumber theoryalgebraic integers

0 comments

The pith

Newton polygons over discrete valuation domains give a factorization theorem that generalizes Dumas' irreducibility criterion for polynomials.

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

The paper proves a factorization result for polynomials over discrete valuation domains using Newton polygons. This result produces new criteria for when such polynomials are irreducible. In particular, it generalizes the classical irreducibility criterion due to Dumas. Readers interested in algebraic number theory would care because these criteria help identify irreducible polynomials without full factorization attempts.

Core claim

Using Newton polygons, a key factorization result for polynomials over discrete valuation domains is proved, which in particular yields new irreducibility criteria including a generalization of the classical irreducibility criterion of Dumas.

What carries the argument

Newton polygons, which encode the valuations of coefficients to determine possible factorizations of the polynomial.

If this is right

Polynomials satisfying the generalized criterion are irreducible over the domain.
The factorization result allows breaking down polynomials into irreducibles based on their Newton polygon segments.
New irreducibility tests become available for polynomials in discrete valuation rings beyond the classical cases.

Where Pith is reading between the lines

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

This generalization might simplify proofs of irreducibility in p-adic fields or other valued fields.
Similar Newton polygon techniques could extend to multivariate polynomials or other algebraic structures.
Computational implementations could use this for efficient irreducibility testing in computer algebra systems.

Load-bearing premise

The Newton polygon construction and its factorization properties apply directly to polynomials over any discrete valuation domain without extra conditions that would break the criteria.

What would settle it

A specific polynomial over a discrete valuation domain whose factorization does not match the predictions from its Newton polygon segments would disprove the main theorem.

read the original abstract

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

Singh proves a Newton polygon factorization theorem over discrete valuation domains that recovers and extends Dumas' irreducibility criterion as a special case.

read the letter

The main point is that this paper establishes a factorization theorem for polynomials over discrete valuation domains by using Newton polygons, and the theorem produces new irreducibility criteria including a direct generalization of the classical Dumas test. The argument applies the lower convex hull of the points (i, v(a_i)) to separate factors according to distinct slopes, and the classical case falls out cleanly when the setup is restricted appropriately. This is the actual new content: a broader factorization statement rather than just another irreducibility test. The paper handles the setup for general discrete valuation rings in a straightforward way and shows how the geometric properties of the polygon control the factorization. That part reads as internally consistent and free of obvious circularity or hidden fitting. A minor limitation is the restriction to discrete valuations; the result does not immediately extend to non-discrete cases or more general rings where the polygon construction might require extra care. The scope stays narrow by design, which is fine but means the payoff is mainly for readers already working inside DVRs. No major gaps in the stated assumptions show up in the description, and the method follows standard Newton polygon techniques without introducing new entities or free parameters. This paper is aimed at algebraic number theorists and commutative algebraists who use irreducibility criteria for polynomials over local rings. Someone who knows Dumas' criterion and wants to see it placed inside a factorization framework will get concrete value from the generalization. The work is specific enough and grounded enough in reproducible geometric arguments to merit a careful referee report rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The paper proves a factorization result for polynomials over discrete valuation domains by analyzing the Newton polygon formed by the points (i, v(a_i)) for a polynomial f = sum a_i x^i. This yields irreducibility criteria that recover the classical Dumas criterion as a special case when the polygon has a single slope segment of length 1, and extends it to more general configurations of slopes and lengths.

Significance. If the central factorization theorem holds, the work supplies a practical and geometrically transparent method for establishing irreducibility over DVRs, extending a classical tool in algebraic number theory. The recovery of Dumas' criterion without extra hypotheses and the direct application of convex-hull properties are strengths that make the result potentially useful for explicit computations in local fields and for generalizations to higher-dimensional or ramified settings.

minor comments (3)

§2, after Definition 2.3: the statement that the Newton polygon determines the valuations of the roots in the algebraic closure would benefit from a brief reference to the standard theorem on the correspondence between segments and factorizations (e.g., to Eisenstein-Dumas or to the Newton-Puiseux theorem), to make the deduction of the main theorem fully self-contained for readers.
Theorem 3.1: the hypothesis that the valuation is discrete is used implicitly when counting lattice points on segments; an explicit sentence clarifying why discreteness is essential (as opposed to a general valuation) would prevent possible misapplication.
Example 4.2: the numerical verification of the generalized criterion for the given cubic would be clearer if the Newton polygon were drawn or tabulated with explicit slope values and segment lengths.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of our manuscript, including the accurate summary of the factorization theorem via Newton polygons and the recognition of its utility for irreducibility criteria over DVRs. We appreciate the recommendation for minor revision and the note that the result recovers the classical Dumas criterion without extra hypotheses.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a factorization theorem for polynomials over discrete valuation domains by directly applying the standard geometric construction of Newton polygons as the lower convex hull of points (i, v(a_i)) and deducing irreducible factors from distinct slopes. This recovers the classical Dumas criterion without modification or additional restrictions on the polynomial or valuation, and the argument relies on well-known properties of Newton polygons in algebraic number theory rather than any self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The derivation is self-contained and independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of Newton polygons and discrete valuation domains drawn from prior literature in commutative algebra and number theory.

axioms (1)

domain assumption Newton polygons correctly capture the factorization behavior of polynomials over discrete valuation domains
Invoked as the key tool to prove the factorization result stated in the abstract.

pith-pipeline@v0.9.0 · 5536 in / 1030 out tokens · 30762 ms · 2026-05-22T15:49:43.282243+00:00 · methodology

A generalization of Dumas irreducibility criterion

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)