{"id":"ae34e52f-9205-4464-8379-6b225c7df401","arxiv_id":"2606.15394","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Provides a Perron-Frobenius matrix realization proving the dominant zero of Nekrasov-Okounkov polynomials is the unique zero of maximal modulus, real, negative, and simple.","lead":"This paper constructs an explicit nonnegative matrix whose eigenvalues are the non-trivial zeros of a family of polynomials including the Nekrasov-Okounkov polynomials, then applies Perron-Frobenius theory to prove the dominant zero is real, negative, and simple. A smart generalist might read it to see how linear algebra tools can characterize zeros of polynomials arising in partition theory and number theory.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"The identification of non-trivial zeros of pol_n^h(-z) with eigenvalues of M_n^h via the Hessenberg determinant representation is the least-secured step.","rationale":"The reader correctly isolated the eigenvalue-identification step as the weakest assumption; the rest of the Perron–Frobenius argument is standard once that step is granted. Because the full manuscript was unavailable to the reader, the verdict remains UNVERDICTED; the concrete test above would resolve the identification without requiring the entire proof.","tokens_in":1820,"tokens_out":427,"duration_ms":32832,"concrete_test":"For n=4 and h(k)=k, compute pol_4^h(z) explicitly from the recurrence, form the monic polynomial p(w) = w^{-1} pol_4^h(-w), construct the 3×3 matrix M_4^h according to the Hessenberg recipe given in the paper, and check whether the eigenvalues of M_4^h coincide with the three nonzero roots of p(w).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The argument first derives a Hessenberg determinant formula for pol_n^h(z), factors out the trivial zero at z=0, and asserts that the remaining roots are precisely the eigenvalues of an explicit nonnegative (n-1)×(n-1) matrix M_n^h. Perron–Frobenius is then applied to this matrix (after proving primitivity) to conclude uniqueness, reality, negativity and simplicity of the dominant zero. If the characteristic polynomial of M_n^h does not exactly equal (up to sign and scaling) the normalized pol_n^h(-z)/z, the entire spectral conclusion fails even if the determinant representation itself is correct. The paper states the identification “follows from” the Hessenberg form, but the explicit construction of the subdiagonal and superdiagonal entries of M_n^h and the verification that they reproduce the recurrence coefficients involving σ(k) and h(n) constitute the single point where an algebraic mismatch would invalidate the claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":2027,"tokens_out":501,"duration_ms":31339,"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":[{"comment":"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.","section":"paragraph following the Hessenberg determinant representation"}],"minor_comments":[{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."}],"tokens_in":1497,"tokens_out":313,"duration_ms":38106,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the authors build an explicit (n-1) by (n-1) nonnegative matrix M_n^h whose eigenvalues are supposed to be exactly the non-trivial zeros of pol_n^h(-z), prove the matrix is primitive, and then invoke Perron-Frobenius to get that the spectral radius is a simple real negative eigenvalue of largest modulus.\n\nWhat stands out as new is the construction of this matrix from the Hessenberg determinant formula for the recursively defined polynomials, followed by the primitivity argument. Prior work on Nekrasov-Okounkov polynomials has looked at their zeros, but this linear-algebraic realization appears fresh.\n\nThe approach works well in turning a recursive definition into a spectral problem where standard theorems apply directly. The specialization to h(n)=n for the Nekrasov-Okounkov case follows immediately.\n\nThe soft spot is the precise identification between the zeros and the eigenvalues. The paper derives the Hessenberg det rep, factors out the zero at origin, and states that the remaining roots match the eigenvalues of M. If the entries of M are chosen so that its characteristic polynomial reproduces the normalized pol_n^h(-z)/z exactly, then everything goes through. But that matching depends on correctly encoding the divisor sum coefficients σ(k) and the h(n) into the matrix entries. A small algebraic slip there would invalidate the PF conclusions even if the det formula is right. The abstract presents it as following directly, but this is the step that needs the most scrutiny in the proofs. The monotonicity of the spectral radii is a nice extra.\n\nThis is for readers already working with Nekrasov-Okounkov polynomials or similar divisor-sum recursions who want tools to locate or bound the dominant zero. It is not a broad advance but a targeted technique.\n\nI would send it to peer review. The core idea is worth checking in detail, and the PF application is clean once the matrix is accepted.","headline":"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.","tokens_in":2538,"tokens_out":496,"would_cite":false,"duration_ms":37670,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"The dominant zero of Nekrasov-Okounkov polynomials is real, negative, and simple.","keywords":["Nekrasov-Okounkov polynomials","dominant zeros","Perron-Frobenius theory","Hessenberg determinant","sum of divisors","primitive matrix","spectral radius"],"falsifier":"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.","tokens_in":2705,"feed_emoji":"","tokens_out":824,"duration_ms":55061,"temperature":0.7,"pith_summary":"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.","feed_headline":"Nekrasov-Okounkov polynomials have unique real negative dominant zero","feed_subtitle":"Perron-Frobenius on an explicit nonnegative matrix from the Hessenberg form locates the zero of largest modulus and proves it is simple.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Unique real negative zero dominates Nekrasov-Okounkov polynomials","Simple negative zero maximizes modulus of Nekrasov-Okounkov polynomials","Perron-Frobenius identifies unique simple dominant zero for Nekrasov-Okounkov","Nekrasov-Okounkov polynomials have simple real negative dominant zero"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The nonzero zeros of pol_n^h(-z) are exactly the eigenvalues of the matrix M_n^h constructed from the Hessenberg determinant representation.","fun_headline_variants_meta":{"raw":{"variants":["Unique real negative zero dominates Nekrasov-Okounkov polynomials","Simple negative zero maximizes modulus of Nekrasov-Okounkov polynomials","Perron-Frobenius identifies unique simple dominant zero for Nekrasov-Okounkov","Nekrasov-Okounkov polynomials have simple real negative dominant zero"]},"model":"grok-4.3","cost_usd":0.00761,"raw_usage":{"total_tokens":3571,"prompt_tokens":839,"num_sources_used":0,"completion_tokens":83,"cost_in_usd_ticks":76099500,"prompt_tokens_details":{"text_tokens":839,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2649,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":839,"tokens_out":83,"duration_ms":32830,"temperature":1.0,"reasoning_tokens":2649,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T04:02:26.425437+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}