arxiv: 2604.25970 · v1 · submitted 2026-04-28 · 🧮 math.GM

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Right edge rates of the zeros of widetilde{Xi}_n and widetilde{Λ}_n

Luc Rams\`es Talla Waffo

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Pith reviewed 2026-05-07 14:02 UTC · model grok-4.3

classification 🧮 math.GM

keywords zeros of polynomialsEulerian polynomialsasymptotic ratesright edgelimiting distributionrescaled polynomialsexponential decay

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The pith

The rescaled polynomials from Eulerian types A and B approach 1 at exponential rates 4^{1-n} and 9^{1-n} despite sharing the same bulk zero distribution.

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

The paper examines the rightmost zeros of two families of rescaled even polynomials. It shows that although the families have the same limiting distribution of all zeros, the largest zero of each family nears 1 at a different exponential speed. For one family the gap to 1 shrinks like 4 to the power 1-n; for the other it shrinks like 9 to the power 1-n. The distinction follows from expressing the polynomials via Eulerian polynomials of type A and type B and then estimating the smallest negative zero from the first non-constant coefficient. Readers care because the result separates global and local zero behavior inside a single family of objects.

Core claim

We prove that the two families have different exponential rates at the right endpoint: 1/(n-1) log(1 - x^{(Lambda)}_{n-1,n}) -> -log 4 and 1/(n-1) log(1 - x^{(Xi)}_{n-1,n}) -> -log 9. Thus, although the two families share the same global limiting zero distribution, their extreme right zeros approach 1 on different exponential scales. The proof rests on the representation of the original polynomials in terms of Eulerian polynomials of type B and type A together with an elementary bound on the smallest negative zero.

What carries the argument

Representation of Xi_n and Lambda_n in terms of Eulerian polynomials of type B and type A, together with an elementary estimate for the smallest negative zero expressed via the first non-constant coefficient.

If this is right

The two families remain distinguishable by their largest zero even when their overall zero distributions coincide.
Explicit exponential approximations become available for the rightmost zero of each family as a function of n.
The same representation-plus-coefficient technique can be applied to other rescaled polynomial sequences that possess Eulerian connections.
Numerical checks for moderate n can be compared directly against the predicted rates without solving the full zero problem.

Where Pith is reading between the lines

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

Edge rates can expose structural differences invisible to the global limiting distribution.
Higher-order corrections to the leading exponential term could be derived by refining the coefficient estimate.
Analogous rate distinctions may appear in other families of polynomials whose coefficients or generating functions involve Eulerian numbers.
Direct numerical verification for n up to several hundred is feasible and would confirm the limits to high precision.

Load-bearing premise

The even polynomials admit exact representations as Eulerian polynomials of type A and type B so that their smallest negative zeros are controlled by the first non-constant coefficient.

What would settle it

For large n compute the largest zero x of each rescaled polynomial and verify whether (n-1) log(1-x) lies near -log 4 for the Lambda family and near -log 9 for the Xi family.

read the original abstract

We consider the two families of even polynomials $\Xi_n$ and $\Lambda_n$ studied in~\cite{TallaWaffo2026arxiv2602.16761}, together with the rescaled polynomials $\widetilde{\Xi}_n(x):=\Xi_n(\sqrt{x})$ and $\widetilde{\Lambda}_n(x):=\Lambda_n(\sqrt{x})$, $n\ge2$. Their zeros are real, simple, and contained in $(0,1)$. Writing them as $0<x^{(\Xi)}_{1,n}<\cdots<x^{(\Xi)}_{n-1,n}<1$ and $0<x^{(\Lambda)}_{1,n}<\cdots<x^{(\Lambda)}_{n-1,n}<1$, we study the asymptotic behaviour of the largest zeros $x^{(\Xi)}_{n-1,n}$ and $x^{(\Lambda)}_{n-1,n}$. We prove that the two families have different exponential rates at the right endpoint: \[ \frac{1}{n-1}\log\bigl(1-x^{(\Lambda)}_{n-1,n}\bigr)\to-\log4, \qquad \frac{1}{n-1}\log\bigl(1-x^{(\Xi)}_{n-1,n}\bigr)\to-\log9. \] Thus, although the two families share the same global limiting zero distribution, their extreme right zeros approach $1$ on different exponential scales. The proof is based on the representation of $\Xi_n$ and $\Lambda_n$ in terms of Eulerian polynomials of type~B and type~A, respectively, and on an elementary estimate for the smallest negative zero in terms of the first non-constant coefficient.

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 gives explicit exponential rates at which the largest zeros of these two rescaled even polynomials approach 1, and shows the rates differ despite identical bulk distributions.

read the letter

The main takeaway is that these two families of even polynomials have the same overall zero distribution in the limit, but their largest zeros get close to 1 at different speeds, with rates given by -log 4 and -log 9 after scaling by 1/(n-1). What the paper does well is reduce the edge question to properties of Eulerian polynomials via the known representations for Lambda_n and Xi_n. The elementary estimate for the smallest negative zero, based on the ratio of coefficients, then directly gives the exponential decay rates for 1 minus the largest zero. This is a nice, low-tech way to separate the edge from the bulk behavior. The soft spot is whether that coefficient ratio estimate is sharp enough for the exact limit. These bounds are often inequalities, and without showing that the error from higher terms vanishes in the scaled log, the constants might not come out precisely. The abstract is brief on this step, so the full write-up needs to confirm there are no lingering factors. This is really for people working on zero asymptotics for polynomials with combinatorial connections. A specialist in that area would get value from seeing how the edge rates differ. It looks like it deserves peer review to verify the details of the proof. I would send it to a referee.

Referee Report

1 major / 0 minor

Summary. The paper proves that the largest zeros of the rescaled polynomials ~Ξ_n and ~Λ_n approach 1 at different exponential rates: 1/(n-1) log(1 - x^(Λ)_{n-1,n}) → -log 4 and 1/(n-1) log(1 - x^(Ξ)_{n-1,n}) → -log 9. The proof relies on expressing Ξ_n and Λ_n in terms of Eulerian polynomials of type B and type A, respectively, together with an elementary estimate for the smallest negative zero in terms of the ratio of the constant term to the linear coefficient.

Significance. If the result holds, it is significant because it shows that two families of polynomials sharing the same global limiting zero distribution can nevertheless have their rightmost zeros approaching the boundary on distinct exponential scales. This refines the understanding of the zero distributions from the referenced work and benefits from the explicit combinatorial link to Eulerian polynomials.

major comments (1)

[Proof of the main result (Eulerian representations and negative-zero estimate)] The proof of the main limits (via the Eulerian representations and the elementary negative-zero estimate): the standard coefficient-ratio bound on the smallest negative zero typically provides only a one-sided estimate or one carrying n-dependent factors. The manuscript must show explicitly that the ratio of the actual zero to this bound tends to 1 (or that higher-degree terms contribute o(1) after taking log and dividing by (n-1)), so that the claimed exact constants -log 4 and -log 9 follow rather than merely bounds or limits with undetermined prefactors. This step is load-bearing for the precise rates asserted.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for identifying the need to confirm the asymptotic sharpness of the negative-zero estimate. We address this point below and will incorporate the requested clarification into the revised manuscript.

read point-by-point responses

Referee: [Proof of the main result (Eulerian representations and negative-zero estimate)] The proof of the main limits (via the Eulerian representations and the elementary negative-zero estimate): the standard coefficient-ratio bound on the smallest negative zero typically provides only a one-sided estimate or one carrying n-dependent factors. The manuscript must show explicitly that the ratio of the actual zero to this bound tends to 1 (or that higher-degree terms contribute o(1) after taking log and dividing by (n-1)), so that the claimed exact constants -log 4 and -log 9 follow rather than merely bounds or limits with undetermined prefactors. This step is load-bearing for the precise rates asserted.

Authors: We agree that the manuscript should explicitly verify the sharpness of the bound to obtain the precise constants. The proof proceeds by first expressing the rescaled polynomials in terms of the Eulerian polynomials of type A and B (whose coefficients are positive and combinatorially explicit), then applying the elementary ratio bound to the constant and linear terms of the transformed polynomial whose negative root determines the rightmost zero of the original family. In the revision we will insert a short lemma establishing that, for these specific families, the higher-degree terms contribute a multiplicative factor 1 + o(1) to the ratio as n grows. This follows from the known generating-function representations and recurrence relations for the Eulerian coefficients, which imply that the summed tail is asymptotically negligible relative to the linear term on the scale relevant to the logarithm. Consequently, log of the bound equals log of the actual zero plus o(1); dividing by (n-1) then yields the exact limits -log 4 and -log 9 without undetermined prefactors. We view this as a clarification rather than a change in the core argument. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain

full rationale

The proof chain rests on an external representation of the rescaled polynomials as Eulerian polynomials of type A/B (cited from prior independent derivation) together with a general coefficient-ratio bound on the smallest negative root. Neither step is self-definitional, fitted to the target rate, nor does any equation reduce the claimed limits to the inputs by construction. The asymptotic extraction is a standard limit argument applied to the explicit leading coefficients and is falsifiable independently of the present paper's fitted values.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the polynomial representations in Eulerian polynomials and the elementary estimate for the smallest negative zero; these are treated as given inputs from prior literature or standard techniques.

axioms (2)

domain assumption Representation of Xi_n and Lambda_n in terms of Eulerian polynomials of type B and type A
Invoked to obtain the zero locations and coefficients needed for the edge analysis
domain assumption Elementary estimate relating the smallest negative zero to the first non-constant coefficient
Used to bound the distance of the largest positive zero to 1 after rescaling

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discussion (0)

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Algebraic representatives of the ratios $\zeta(2n+1)/\pi^{2n}$ and $\beta(2n)/\pi^{2n-1}$

[1] Luc Ramsès Talla Waffo. Algebraic representatives of the ratios ζ(2n + 1) /π 2n and β(2n)/π 2n−1. arXiv preprint. Submitted on 18 Feb 2026. 2026. arXiv: 2602.16761 [math.NT] . URL: https://arxiv.org/ abs/2602.16761. 5

work page internal anchor Pith review Pith/arXiv arXiv 2026