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arxiv: 2602.15718 · v2 · submitted 2026-02-17 · 🧮 math.CA · math.CO· math.NT

Recognition: no theorem link

Asymptotics and zero behaviour of geometric polynomials

M. Bello-Hern\'andez , M. Benito , \'O. Ciaurri , E. Fern\'andez
Authors on Pith no claims yet

Pith reviewed 2026-05-15 22:01 UTC · model grok-4.3

classification 🧮 math.CA math.COmath.NT
keywords geometric polynomialsEulerian polynomialsasymptoticszero distributionorthogonalitycomplex planeinterval analysis
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The pith

Geometric polynomials admit explicit asymptotics in the cut plane and regularly spaced zeros inside (-1,0), with all statements transferring directly to Eulerian polynomials.

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

The paper derives asymptotic expansions for geometric polynomials that hold uniformly in the complex plane outside the segment [-1,0] and also on the open interval (-1,0). It supplies an explicit formula for the distance between consecutive zeros in the interior of (-1,0). Orthogonality relations satisfied by the geometric polynomials are established as well. Because the two families are linked by a simple algebraic identity, every one of these statements applies verbatim to the Eulerian polynomials.

Core claim

Geometric polynomials possess asymptotic representations valid throughout the complex plane minus the cut [-1,0] and on the interval (-1,0); the distance between successive zeros approaches a definite limiting value in the bulk of (-1,0); and the polynomials obey a family of orthogonality identities. All three classes of results carry over unchanged to Eulerian polynomials by virtue of the algebraic relation connecting the two sequences.

What carries the argument

The algebraic relation between geometric polynomials and Eulerian polynomials, which transmits every asymptotic formula, zero-spacing law, and orthogonality identity without adjustment.

If this is right

  • Asymptotic expansions exist uniformly away from the cut [-1,0].
  • Consecutive zeros inside (-1,0) are separated by a distance that approaches a constant in the bulk.
  • Geometric polynomials satisfy explicit orthogonality relations.
  • All of the above statements hold verbatim for Eulerian polynomials.

Where Pith is reading between the lines

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

  • The zero-spacing result supplies a practical way to locate roots of Eulerian polynomials without solving the defining recurrence for each degree.
  • The exterior asymptotics can be used to estimate the radius of convergence of generating functions built from these polynomials.
  • Similar spacing laws may hold for other polynomial families connected to Eulerian numbers by comparable generating-function identities.

Load-bearing premise

The algebraic link between geometric and Eulerian polynomials is strong enough that every derived asymptotic, spacing, and orthogonality statement transfers exactly as written.

What would settle it

A direct numerical check for large degree showing that the distance between consecutive zeros near the middle of (-1,0) deviates from the predicted asymptotic spacing, or that the error in the asymptotic formula at a fixed exterior point fails to tend to zero.

Figures

Figures reproduced from arXiv: 2602.15718 by E. Fern\'andez, M. Bello-Hern\'andez, M. Benito, \'O. Ciaurri.

Figure 1
Figure 1. Figure 1: Graphics of Pn(z) for n = 0, 1, . . . , 6 in the interval [−1.01, 0.1]. Proof. 1. As  n + 1 k  = k  n k  +  n k − 1  , from (2) we obtain Pn(z) = nX−1 k=1 k!k  n − 1 k  z k + Xn k=2 k!  n − 1 k − 1  z k = z dPn−1(z) dz + z d(zPn−1(z)) dz , and (9) follows. The statement about the zeros of P1(z) = z is trivial. From (9) and since (1 + z)Pn−1(z) is zero at z = −1 and at the zeros of Pn−1(z) by Roll… view at source ↗
Figure 2
Figure 2. Figure 2: Level curve of the modulus of Pn(z)(log 1+z z ) n+1 n! (left) and its approximation, 1 1+z (right) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hue of the arguments for Pn(z)(log 1+z z ) n+1 n! (left) and its approximation, 1 1+z (right). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graphics of (z+1)P25(z)((ℓ(z))2+π 2 ) 13 25! (left) and its approximation, (ℓ(z)+πi) 26+(ℓ(z)−πi) 26 ((ℓ(z))2+π2) 13 (right). Theorem 5. There exists a sequence (wn)n∈N ⊂ (−1, 0), limn→∞ wn = −1 such that z(1 + z) 2P ′ n (z)((ℓ(z))2 + π 2 ) (n+2)/2 n! n − Jn+1(z) = o( 1 n ) uniformly for z ∈ [wn, −1 − wn] as n → ∞. The following results are analogous of the previous ones for Eulerian polynomials. Theorem 6… view at source ↗
Figure 5
Figure 5. Figure 5: The polynomial P50 and, in red, the approximation of its zeros (on the left). The polynomial P55 and, in red, the approximation of its zeros (on the right). We are then in a position to prove the clock behavior of the zeros of Pn(z) in the bulk of (−1, 0) of Theorem 2. Proof of Theorem 2. Let us check only when n is even, the other case is similar. Ac￾cording to Lemma 8 the zeros ηk,n of Jn(z) lie in (−1, … view at source ↗
read the original abstract

We obtain some results on the asymptotic behaviour of Geometric polynomials in both the complex plane minus $[-1,0]$ and the interval $(-1,0)$. We also find the distance of consecutive zeros of these polynomials in the bulk of the interval $(-1,0)$. We also prove that they satisfy certain orthogonality properties. Due to their relationship with Eulerian polynomials, the results for Geometric polynomials can be transferred to Eulerian polynomials verbatim.

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

2 major / 2 minor

Summary. The manuscript obtains asymptotic results for geometric polynomials in the complex plane excluding the cut [-1,0] and on the interval (-1,0), determines the spacing of consecutive zeros in the bulk of (-1,0), establishes certain orthogonality relations, and asserts that all of these results transfer verbatim to Eulerian polynomials via their known relationship.

Significance. If the derivations are complete and the transfer is justified without additional correction terms, the work would supply concrete asymptotic expansions and zero-spacing formulas for two combinatorially important polynomial families, potentially strengthening links between analytic combinatorics and orthogonal-polynomial techniques.

major comments (2)
  1. [Relationship section] The section establishing the relationship between geometric and Eulerian polynomials: the claim that zero spacings and asymptotic expressions transfer verbatim is not accompanied by an explicit change-of-variable formula or Jacobian verification. If the standard relation involves a non-linear map (e.g., a Möbius transformation sending (-1,0) to another interval), consecutive-zero distances transform by the absolute value of the derivative; the bulk-spacing result therefore requires an explicit multiplicative factor rather than verbatim passage.
  2. [Zero behaviour on (-1,0)] The derivation of the zero-spacing formula in the bulk of (-1,0): no error estimate or uniformity statement is supplied for the distance between consecutive zeros, making it impossible to confirm that the claimed asymptotic holds uniformly away from the endpoints.
minor comments (2)
  1. [Introduction] Notation for the geometric polynomials is introduced without a displayed definition or reference to the standard generating function; a single displayed equation would remove ambiguity.
  2. [Orthogonality properties] The orthogonality statement is phrased only qualitatively; the precise inner-product measure and the range of indices should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments identify two places where additional explicit details will strengthen the manuscript. We address each point below and will revise accordingly.

read point-by-point responses

  1. Referee: [Relationship section] The section establishing the relationship between geometric and Eulerian polynomials: the claim that zero spacings and asymptotic expressions transfer verbatim is not accompanied by an explicit change-of-variable formula or Jacobian verification. If the standard relation involves a non-linear map (e.g., a Möbius transformation sending (-1,0) to another interval), consecutive-zero distances transform by the absolute value of the derivative; the bulk-spacing result therefore requires an explicit multiplicative factor rather than verbatim passage.

    Authors: We agree that the transfer requires an explicit change-of-variable formula and verification of how distances transform. The relationship between the two families is given by a standard substitution that maps the interval (-1,0) to the corresponding interval for the Eulerian polynomials. In the revision we will insert the precise mapping, compute the absolute value of its derivative, and multiply the bulk-spacing formula by this factor so that the passage is fully justified rather than stated as verbatim. revision: yes

  2. Referee: [Zero behaviour on (-1,0)] The derivation of the zero-spacing formula in the bulk of (-1,0): no error estimate or uniformity statement is supplied for the distance between consecutive zeros, making it impossible to confirm that the claimed asymptotic holds uniformly away from the endpoints.

    Authors: The spacing formula follows from the leading asymptotic expansion together with the orthogonality relation that controls the oscillatory behavior in the bulk. We acknowledge that an explicit error bound and uniformity statement are missing. In the revised manuscript we will add a uniformity claim: for any fixed compact subinterval [a,b] subset (-1,0) the distance between consecutive zeros satisfies the stated asymptotic with an error of O(1/n) uniformly on that subinterval, derived from the remainder in the asymptotic expansion away from the endpoints. revision: yes

Circularity Check

0 steps flagged

No circularity in derivation chain for asymptotics, zeros, or transfer

full rationale

The paper derives asymptotic behavior, consecutive zero distances in the bulk, and orthogonality for geometric polynomials independently, then transfers the results verbatim to Eulerian polynomials via a pre-existing relationship. No steps reduce by construction to inputs, no parameters are fitted and relabeled as predictions, and no load-bearing self-citations or uniqueness theorems from the same authors are invoked. The transfer relies on an external known connection between the families rather than any self-referential definition or ansatz smuggling. The chain is self-contained against standard analytic methods for such polynomials.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no free parameters, axioms, or invented entities are identified or required for the stated claims.

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Reference graph

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