arxiv: 2605.07628 · v1 · submitted 2026-05-08 · 🧮 math.CV · math.OC

Recognition: 2 theorem links

· Lean Theorem

The idealizer of the set of quasi-stable polynomials

Micha{\l} Kudra

Pith reviewed 2026-05-11 01:56 UTC · model grok-4.3

classification 🧮 math.CV math.OC

keywords stable polynomialsquasi-stable polynomialsHadamard productidealizersemigrouppolynomial coefficients

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The pith

The idealizer of the semigroup of stable polynomials is the set of quasi-stable polynomials

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

The paper examines the idealizer of the set of stable polynomials of degree n under the Hadamard product, which is the largest subsemigroup where the stable polynomials form an ideal. It proposes a conjecture that this idealizer is precisely the quasi-stable polynomials. This is proved for all n up to 5. The necessity of the quasi-stable condition is established for general n, while sufficiency is shown in a special case. A sympathetic reader would care because this gives a concrete algebraic description of how stable polynomials interact with others in the semigroup.

Core claim

It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree n, denoted by H_n, forms an abelian semigroup under the Hadamard product inside the abelian group R_n^+ of degree-n polynomials with positive real coefficients. The idealizer of H_n is the largest subsemigroup of R_n^+ in which H_n is an ideal. The paper conjectures that the idealizer consists exactly of the quasi-stable polynomials, proves this for n ≤ 5, shows that the condition is necessary in general, and establishes sufficiency in a distinguished special case.

What carries the argument

The idealizer of H_n, defined as the largest subsemigroup of R_n^+ making H_n an ideal under the Hadamard product.

Load-bearing premise

The quasi-stable condition exactly identifies which polynomials belong to the idealizer of H_n for every n.

What would settle it

A polynomial of degree 6 that is quasi-stable yet fails to keep the product with every stable polynomial inside the stable set, or a non-quasi-stable polynomial that does keep all such products inside the idealizer.

read the original abstract

It follows from the Garloff-Wagner Theorem that the set of stable polynomials of degree $n$, denoted by $\mathcal{H}_n$, i.e., those whose zeros all lie in the open left complex half-plane, with the Hadamard product $*$, forms an abelian semigroup contained in the abelian group $\mathbb{R}_n^+$ of polynomials of degree $n$ with positive real coefficients. By the idealizer of the set $\mathcal{H}_n$, we refer to the largest subsemigroup of $\mathbb{R}_n^+$ in which $\mathcal{H}_n$ is an ideal. In this paper, we formulate a conjecture characterizing the idealizer of $\mathcal{H}_n$ and prove it for $n \leqslant 5$. In addition, we show that the proposed condition is necessary for any polynomial to belong to the idealizer and establish, in a distinguished special case, a sufficient condition of a similar nature that supports the conjecture.

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

The paper conjectures that quasi-stable polynomials form the idealizer of stable polynomials under the Hadamard product, with a general necessity proof and verification for n up to 5.

read the letter

The paper's main contribution is a conjecture that the idealizer of the set of degree-n stable polynomials under the Hadamard product is exactly the set of quasi-stable polynomials. They prove that any polynomial in the idealizer must be quasi-stable, verify the full characterization for degrees up to 5, and give a sufficient condition in a special case. This builds directly on the Garloff-Wagner theorem by identifying the largest semigroup where H_n is an ideal. The necessity argument stands on its own and applies for arbitrary n, which is a solid piece of work. The explicit proofs for n ≤ 5 provide verifiable checks that support the idea. The soft spot is that the conjecture remains open for n > 5. Sufficiency holds only in one distinguished special case rather than in general, so the extension to all degrees rests on the unproven part. No search for counterexamples or additional supporting arguments for larger n appears in the work. Readers working on polynomial stability, Hadamard products, or related semigroups in complex analysis would find the necessity result and the small-degree verifications of interest. It is a narrow but precise algebraic question, so the audience is specialized. The paper engages honestly with the literature and separates what is proven from what is conjectured. It deserves a serious referee to examine the small-n proofs in detail and to assess whether the conjecture can be settled or if counterexamples exist for higher degrees. I would recommend sending it out for peer review.

Referee Report

2 major / 0 minor

Summary. The paper claims that the idealizer of the set H_n of stable polynomials of degree n (under Hadamard product) within the group of degree-n polynomials with positive real coefficients is precisely the set of quasi-stable polynomials. It formulates this characterization as a conjecture, proves that the quasi-stable condition is necessary for any polynomial to lie in the idealizer (for arbitrary n), establishes the full conjecture for n ≤ 5, and proves a sufficient condition of similar nature in one distinguished special case.

Significance. If the conjecture holds, the work would supply a complete algebraic characterization of the largest subsemigroup of R_n^+ in which H_n forms an ideal, thereby extending the Garloff-Wagner theorem. The general necessity proof and the complete resolution for n ≤ 5 are concrete, self-contained contributions; the special-case sufficiency supplies supporting evidence. The open status for n > 5, however, means the full impact remains conditional on future resolution of the conjecture.

major comments (2)

The central conjecture asserts that the idealizer equals the quasi-stable polynomials for every n, yet necessity is shown in general while sufficiency is established only for n ≤ 5 and in one distinguished special case. This gap is load-bearing for the claimed characterization of the idealizer.
No general argument, reduction, or counterexample search for n > 5 is supplied, so the extension of the characterization beyond the verified range rests entirely on the unproven conjecture.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for acknowledging the necessity proof and the results for n ≤ 5. We respond to the major comments point by point.

read point-by-point responses

Referee: The central conjecture asserts that the idealizer equals the quasi-stable polynomials for every n, yet necessity is shown in general while sufficiency is established only for n ≤ 5 and in one distinguished special case. This gap is load-bearing for the claimed characterization of the idealizer.

Authors: The manuscript formulates the equality as a conjecture rather than a theorem. It proves necessity for arbitrary n, sufficiency for all n ≤ 5, and a related sufficient condition in one special case. The gap for n > 5 is stated explicitly as part of the conjecture. The paper therefore does not claim a complete characterization beyond the proven cases, and the presentation already reflects the conditional nature of the result. revision: no
Referee: No general argument, reduction, or counterexample search for n > 5 is supplied, so the extension of the characterization beyond the verified range rests entirely on the unproven conjecture.

Authors: This observation is correct: the manuscript supplies neither a general proof nor a counterexample search for n > 5. That is precisely why the statement is left as a conjecture. The contributions consist of the general necessity result, the complete verification for n ≤ 5, and the supporting special-case sufficiency. These are self-contained and extend the Garloff-Wagner theorem in the directions achieved. revision: no

Circularity Check

0 steps flagged

No circularity: conjecture with independent necessity proof and small-n verification

full rationale

The paper formulates a new conjecture that the idealizer of H_n equals the quasi-stable polynomials, proves the proposed condition is necessary for any n, establishes the full characterization for n ≤ 5, and gives a sufficient condition only in one special case. It invokes the external Garloff-Wagner Theorem solely as background for the semigroup structure of H_n; this prior result is independent and not load-bearing for the new conjecture or its partial proofs. No self-definitional steps, fitted inputs renamed as predictions, self-citation chains, or ansatz smuggling occur. The derivation chain is self-contained against external benchmarks and does not reduce the central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract invokes the Garloff-Wagner Theorem as the starting point for the semigroup structure but introduces no free parameters, no new invented entities, and no additional ad-hoc axioms beyond standard algebraic definitions.

axioms (1)

standard math Garloff-Wagner Theorem: the set of stable polynomials of degree n with positive real coefficients forms an abelian semigroup under the Hadamard product
Cited as the background fact that H_n is a semigroup inside R_n^+.

pith-pipeline@v0.9.0 · 5458 in / 1394 out tokens · 45762 ms · 2026-05-11T01:56:31.626553+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
Conjecture 1. For any n∈N, equality Y_n = X_n holds... We prove that Conjecture 1 is true for n=4 (Theorem 8) and n=5 (Theorem 9).
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
Y_n := {g∈R_n^+ : g∗Q_k^m / x^m ∈ H^*_k ∀k,m∈N, k≥2, k+m≤n}

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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