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Dominant Zeros of Nekrasov--Okounkov Polynomials

T0 review · 1 major / 1 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read The dominant zero of Nekrasov-Okounkov polynomials is real, negative, and simple.

desk verdict The paper builds an explicit nonnegative matrix from the Hessenberg form whose eigenvalues match the non-trivial zeros, then uses primitivity plus Perron-Frobenius to pin down the dominant one as unique, real, negative, and simple. read the letter →

arxiv 2606.15394 v2 pith:4IGMZKPO submitted 2026-06-13 math.CO math.NT

Bernhard Heim , Markus Neuhauser , with an appendix by Ken Ono This is my paper · ORCID
classification math.COmath.NT
keywords Nekrasov-OkounkovpolynomialsdominantzerosPerron-FrobeniustheoryHessenbergdeterminantsumofdivisorsprimitivematrixspectralradius
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper defines a recursive family of polynomials pol_n^h(z) using the sum-of-divisors function and a positive sequence h. It produces a Hessenberg determinant form for these polynomials and shows that their nonzero zeros of pol_n^h(-z) are the eigenvalues of an explicit nonnegative matrix of size n-1. Perron-Frobenius theory is applied after proving the matrix is primitive, yielding a unique real negative simple zero of largest modulus. The Nekrasov-Okounkov polynomials, obtained by setting h(n) equal to n and shifting the argument by 1, inherit the same property. The result also includes strict monotonicity for the spectral radii of the matrices.

What carries the argument

The (n-1) by (n-1) nonnegative primitive matrix M_n^h whose eigenvalues are identified with the nonzero zeros of pol_n^h(-z), so that its Perron eigenvalue determines the dominant zero.

What would settle it

An explicit calculation for some small n greater than 2 in which the characteristic polynomial of M_n^h has a root whose modulus exceeds that of the largest-modulus zero of the corresponding pol_n^h(-z), or in which two distinct zeros share the maximal modulus.

Watch

Extended reading notes

Core claim

For a normalized positive sequence h with h(1)=1, the polynomials satisfy the recursion pol_n^h(z) = (z / h(n)) sum_{k=1}^n sigma(k) pol_{n-k}^h(z). They admit a Hessenberg determinant representation. After removing the trivial zero at the origin, the remaining zeros of pol_n^h(-z) are the eigenvalues of an explicit (n-1) by (n-1) nonnegative matrix M_n^h. This matrix is primitive. Perron-Frobenius theory therefore implies that pol_n^h(z) has a unique zero of maximal modulus; this zero is real, negative, and simple. The same holds for the Nekrasov-Okounkov polynomials nop_n(z) = pol_n^h(z+1) when h(n)=n. The associated spectral radii are strictly monotone.

Load-bearing premise

The nonzero zeros of pol_n^h(-z) are exactly the eigenvalues of the matrix M_n^h constructed from the Hessenberg determinant representation.

Editorial extensions

If this is right

  • pol_n^h(z) has a unique zero of maximal modulus that is real, negative, and simple.
  • The Nekrasov-Okounkov polynomials nop_n(z) have a unique dominant zero that is real, negative, and simple.
  • The spectral radii of the matrices M_n^h are strictly monotone as n increases.
  • The location of the dominant zero is controlled by the Perron eigenvalue of an explicitly constructible matrix.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The matrix construction supplies a practical method to compute the dominant zero numerically for moderate n by standard eigenvalue routines.
  • The same Perron-Frobenius argument may apply to other recursive families whose coefficients involve the divisor function.
  • The strict monotonicity of the spectral radii gives a lower bound on how fast the dominant zero moves with n.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper defines polynomials pol_n^h(z) recursively using the sum-of-divisors function σ(k) for a positive sequence h with h(1)=1, derives a Hessenberg determinant representation, factors out the trivial zero at z=0, identifies the remaining zeros of pol_n^h(-z) with the eigenvalues of an explicit nonnegative (n-1)×(n-1) matrix M_n^h, proves M_n^h is primitive, and applies Perron-Frobenius theory to conclude that pol_n^h(z) has a unique zero of maximal modulus that is real, negative, and simple. The same conclusion is transferred to the Nekrasov-Okounkov polynomials via the specialization h(n)=n, along with a proof of strict monotonicity of the associated spectral radii.

Significance. If the eigenvalue identification holds, the work supplies an exact finite-dimensional nonnegative matrix realization of the dominant zero together with a primitivity proof, yielding a clean Perron-Frobenius argument for uniqueness, reality, negativity and simplicity. This constitutes a concrete advance in the analytic combinatorics of divisor-sum recurrences and supplies falsifiable predictions for the location of the dominant zero that can be checked numerically for small n.

major comments (1)
  1. [paragraph following the Hessenberg determinant representation] The identification that the non-trivial zeros of pol_n^h(-z) are exactly the eigenvalues of M_n^h (stated immediately after the Hessenberg determinant representation) is load-bearing for the entire Perron-Frobenius conclusion. The manuscript must exhibit the explicit subdiagonal and superdiagonal entries of M_n^h in terms of σ(k) and h(n) and verify that the characteristic polynomial of M_n^h equals (up to sign and scaling) the normalized polynomial pol_n^h(-z)/z; an algebraic mismatch would invalidate the spectral claim even if the determinant formula itself is correct.
minor comments (1)
  1. [Abstract] Notation: the shift relating pol_n^h and the Nekrasov-Okounkov polynomials is written nop_n(z) = pol_n^h(z+1); a short sentence clarifying the precise normalization of nop_n would help readers who consult only the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater explicitness in the matrix identification. We address the major comment below.

read point-by-point responses

  1. Referee: [paragraph following the Hessenberg determinant representation] The identification that the non-trivial zeros of pol_n^h(-z) are exactly the eigenvalues of M_n^h (stated immediately after the Hessenberg determinant representation) is load-bearing for the entire Perron-Frobenius conclusion. The manuscript must exhibit the explicit subdiagonal and superdiagonal entries of M_n^h in terms of σ(k) and h(n) and verify that the characteristic polynomial of M_n^h equals (up to sign and scaling) the normalized polynomial pol_n^h(-z)/z; an algebraic mismatch would invalidate the spectral claim even if the determinant formula itself is correct.

    Authors: We agree that the current presentation would benefit from an explicit display of the matrix entries and a direct verification of the characteristic polynomial. In the revised manuscript we will define M_n^h by stating its subdiagonal and superdiagonal entries explicitly in terms of σ(k) and h(n), and we will include a short argument (derived from the already-established Hessenberg determinant representation) confirming that the characteristic polynomial of M_n^h coincides, up to sign and scaling, with the normalized polynomial pol_n^h(-z)/z. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation is self-contained via recurrence to matrix

full rationale

The paper starts from the explicit linear recurrence for pol_n^h(z) involving σ(k) and h(n), derives a Hessenberg determinant representation, factors out the trivial zero, and constructs an explicit nonnegative matrix M_n^h whose characteristic polynomial matches the normalized polynomial by the determinant identity. Perron-Frobenius is then applied to the matrix properties (primitivity proved directly). This is standard linear algebra converting a recurrence into a companion/Hessenberg matrix; the eigenvalue identification is algebraic, not tautological or fitted. No self-citations, no parameter fitting, and no ansatz smuggled in. The result is externally verifiable from the recurrence alone.

Assumptions & free parameters 0 free parameters · 1 assumptions · 1 invented entities

The central claim rests on the standard Perron-Frobenius theorem and the construction of the matrix from the polynomial's recursive definition and Hessenberg form.

assumptions (1)
  • standard math Perron-Frobenius theorem for primitive nonnegative matrices
    Invoked to conclude the existence of a unique positive real eigenvalue of maximal modulus.
invented entities (1)
  • Matrix M_n^h
    purpose: To realize the zeros of the polynomial as its eigenvalues
    Constructed from the recursive definition and Hessenberg form; no external evidence provided beyond the paper's derivation.

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Cite this review

Pith. "Pith review of Dominant Zeros of Nekrasov--Okounkov Polynomials." pith.science (2026). https://pith.science/paper/4IGMZKPO

@misc{pith2026260615394,
  author       = {Pith},
  title        = {Pith review of: Dominant Zeros of Nekrasov--Okounkov Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IGMZKPO}},
  note         = {Machine review of arXiv:2606.15394}
}
abstract

We give an exact finite-dimensional Perron--Frobenius realization of the dominant zero of the Nekrasov--Okounkov polynomials $\nop _n(z)$. For a normalized positive sequence $h=(h(n))_{n\ge 1}$ with $h(1)=1$, define $\pol _0^h(z)=1$ and, for $n\ge 1$, \[ \pol _n^h(z)=\frac{z}{h(n)}\sum_{k=1}^n \sigma(k)\pol _{n-k}^h(z),\] where $\sigma(k)$ denotes the sum of divisors of $k$. The Nekrasov--Okounkov polynomials are obtained from the specialization $h(n)=n$ by the shift $\nop _n(z)=\pol _n^h(z+1)$. We derive a Hessenberg determinant representation for $\pol _n^h(z)$. After separating the trivial zero at the origin, the remaining zeros of $\pol _n^h(-z)$ are identified with the eigenvalues of an explicit $(n-1)\times(n-1)$ nonnegative matrix $M_n^h$. We prove that $M_n^h$ is primitive and apply Perron--Frobenius theory to show that $\pol _n^h(z)$ has a unique zero of maximal modulus; this zero is real, negative, and simple. As a consequence, the same property holds for the Nekrasov--Okounkov polynomials. We also prove strict monotonicity of the associated spectral radii.

Discussion (0). Continue with ORCID to comment.

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