Real zeros of random cosine polynomials with palindromic blocks of coefficients

Ali Pirhadi

REVIEW 3 major objections 3 minor 33 references

Real zeros of random cosine polynomials with palindromic blocks of coefficients

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Random cosine polynomials with coefficients arranged in palindromic blocks of fixed length $\ell$ have expected real-zero count $\frac{2n}{\sqrt3}K_\ell+O(n^{2/3})$, with $K_\ell>1$ for $\ell\ge2$.

desk verdict New explicit constant for palindromic-block cosine polynomials, but Theorem 2.1 overclaims: false at ℓ=1 and only proved on a parity subsequence. read the letter →

arxiv 1908.08154 v1 pith:YH54PMRE submitted 2019-08-22 math.PR

Ali Pirhadi This is my paper · ORCID

classification math.PR MSC 30C1526C10

keywords randomcosinepolynomialsexpectedrealzerospalindromicblocksdependentcoefficientsKac-RiceformulaGaussianasymptoticexpansion

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

This paper studies a random cosine polynomial $V_n(x)=\sum_{j=0}^n a_j\cos(jx)$ whose Gaussian coefficients are no longer independent: they are grouped into palindromic blocks of a fixed length $\ell$, so each block is repeated in reverse order on the other side of the coefficient sequence. The central result is that the expected number of real zeros in $(0,2\pi)$ is $\frac{2n}{\sqrt3}K_\ell+O(n^{2/3})$, where $K_\ell$ is an explicit double integral that depends only on $\ell$ and is strictly larger than 1 for every $\ell\ge2$. This means forcing palindromic-block correlation into the coefficients strictly increases the expected zero count compared with the classical i.i.d. case, by about 6.4% for $\ell=2$ and by smaller amounts as $\ell$ grows. The paper makes the constant $K_\ell$ computable, so the prediction is concrete and testable for each block length.

What carries the argument

The argument runs through the Kac-Rice formula, which expresses the expected number of real zeros of a Gaussian random function as $\frac1\pi\int \sqrt{AC-B^2}/A\,dx$ in terms of the coefficient-level sums $A,B,C$. The key object is the block-ratio function $u_\ell(s)=\sin(\ell s)/(\ell\sin s)$, which encodes the palindromic-block structure: after trigonometric simplification, the quantities $A(x)$, $B(x)$, and $C(x)$ for the block polynomial are approximated by $n$-scale expressions built from $u_\ell$, leaving the integrand $\sqrt{1+\frac{3(1-u_\ell(x)^2)}{(1+u_\ell(x)\cos(nx))^2}}$ on most of $[0,\pi/2]$. The final step replaces the rapidly oscillating $\cos(nx)$ by a Riemann sum over intervals of length $\pi/n$, using a modified-sawtooth error estimate, to obtain the double-integral constant $K_\ell$; the remaining terms are controlled by comparing with the known behavior of i.i.d. cosine polynomials near the endpoints.

What would settle it

For a fixed $\ell$ (say $\ell=2$), evaluate the Kac-Rice integral numerically for $n$ with $n-\ell$ odd and with $n-\ell$ even, and compare both against $(2n/\sqrt3)K_2+O(n^{2/3})$; if the even-parity sequence differs by more than the stated error, the theorem's claim for all $r$ fails. Independently, for $\ell=1$, count all real zeros of the palindromic-coefficient polynomial; if the leading term is $n+n/\sqrt3$ rather than $n/\sqrt3$, the theorem's $\ell=1$ reading omits the deterministic zeros.

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Extended reading notes

Core claim

For a fixed block length $\ell$, write $n=2\ell m+r$ with $m\in\mathbb N$ and $r\in\{-1,\dots,2\ell-2\}$, and let the coefficient vector consist of $2m$ blocks of length $\ell$ arranged so that the first $m$ blocks are repeated in reverse order by the last $m$ blocks, with the leftover $r+1$ coefficients i.i.d. Gaussian. The paper's Theorem 2.1 asserts that the expected number of real zeros of $V_n$ in $(0,2\pi)$ satisfies $E[N_n(0,2\pi)] = \frac{2n}{\sqrt3}K_\ell + O(n^{2/3})$, where $K_\ell=\frac{1}{\pi^2}\int_0^\pi\int_0^{\pi/2}\sqrt{1+\frac{3(1-u_\ell(s)^2)}{(1+u_\ell(s)\cos t)^2}}\,ds\,dt$ and $u_\ell(s)=\frac{\sin(\ell s)}{\ell\sin s}$. The constant is strictly greater than 1 for $\ell\ge2$, so palindromic blocks produce more expected real zeros than independent coefficients, and the size of the excess is read off from $K_\ell$. For $\ell=1$ the formula gives $K_1=1/2$ and a leading term $n/\sqrt3$, which the paper connects to the earlier palindromic-coefficient result, with the caveat that the fully palindromic case carries $n$ additional deterministic zeros recorded in equation (1).

Load-bearing premise

The proof of the main theorem assumes $n-\ell$ is odd and never supplies an argument for the opposite parity, while the theorem is stated for all $n=2\ell m+r$; the $\ell=1$ case also drops the $n$ deterministic zeros that the paper itself acknowledges in equation (1).

Editorial extensions

If this is right

For every fixed $\ell\ge2$, the expected zero count of these dependent polynomials is asymptotically $K_\ell$ times the classical $2n/\sqrt3$, with $K_\ell>1$; for $\ell=2$ and $\ell=3$ the numerical constants are about 1.0642 and 1.0408.
The constant $K_\ell$ is explicitly computable as a double integral, so the predicted excess can be compared with simulations or with exact Kac-Rice evaluations for any chosen block length.
If the theorem is correct, the classical $2n/\sqrt3$ is not just the i.i.d. answer but a lower benchmark for this family of dependent structures: palindromic blocks raise it, while earlier pairwise-equal-block constructions keep it unchanged.
The factor $K_\ell$ decreases toward 1 as $\ell$ grows (the table gives $K_{2019}\approx1.000046$), meaning long palindromic blocks behave almost like the independent-coefficient case.

Reading between the lines

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

The theorem's $\ell=1$ special case, taken literally, gives a leading term $n/\sqrt3$ for fully palindromic coefficients, but the paper's own equation (1) says the actual expected count is $n+n/\sqrt3$ because of deterministic zeros; a unified reading is that the double-integral formula captures only the random zeros, and a version covering $\ell=1$ would need an extra deterministic $n$ term.
Because the proof assumes $n-\ell$ is odd, the most direct check of the full theorem is to test the even-parity subsequence numerically; if it shows the same constant $K_\ell$, a symmetric argument for the missing parity likely exists, and if not, the theorem as stated is too strong.
The numerical decrease of $K_\ell$ toward 1 suggests a quantitative conjecture outside the paper: $K_\ell-1$ should decay like a power of $1/\ell$, and its leading exponent could be extracted from the double-integral formula.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The real news: for ℓ≥2 the paper derives a concrete double-integral constant Kℓ and shows it exceeds 1, so palindromic blocks genuinely increase the expected zero density over the i.i.d. case. That is a new, citable benchmark, and the proof machinery — Riemann-sum approximation plus bounded-variation estimates for oscillatory integrands — is substantive and looks coherent for the regime it actually covers. No fitted parameters; the table is honest numerical evaluation. The two lemmas imported from the author's preprint are auxiliary estimates, so the self-citation is not a problem.

The soft spots are in the gap between the theorem statement and the proof. Equation (1) in the introduction correctly says palindromic coefficients (ℓ=1) have n deterministic zeros, so the expected count is n + n/√3, not n/√3. But Theorem 2.1 with K1=1/2 gives n/√3, and the proof even says "allowing ℓ=1 gives us n/√3." That is a direct contradiction, not a minor typo. The theorem should either exclude ℓ=1 or handle the deterministic-factor case separately.

For ℓ≥2, the proof assumes n−ℓ is odd to get the π/2 symmetry that turns E[N(0,π)] into 2E[N(0,π/2)], and later assumes n is even for the Riemann-sum split. For even ℓ those two assumptions cannot both hold; for odd ℓ they restrict to one parity of n. The theorem states a full O(n^{2/3}) asymptotic for all allowed n, but the proof never addresses the complementary parity. The "for simplicity" sentences acknowledge the restrictions in passing but never reconcile them with the stated result.

None of this kills the likely truth of the ℓ≥2, covered-parity theorem. But as written, the central claim is not supported, and the ℓ=1 case is false. The paper deserves a serious referee — it is not a desk reject — but the referee should send it back for major revision: fix or exclude ℓ=1, and either prove the missing parity or restrict the theorem to the parity actually treated.

Referee Report

3 major / 3 minor

Summary. The paper studies the expected number of real zeros in (0,2π) of a random cosine polynomial V_n(x)=∑_{j=0}^n a_j cos(jx) whose Gaussian coefficients are arranged into palindromic blocks of fixed length ℓ. The main result, Theorem 2.1, claims that for every fixed ℓ and every n=2ℓm+r in the stated range, E[N_n(0,2π)] = (2n/√3) K_ℓ + O(n^{2/3}), where K_ℓ is an explicit double integral, K_ℓ>1 for ℓ≥2, and K_1=1/2. The proof uses the Kac–Rice formula, asymptotic expansions of A(x), B(x), C(x), a Riemann-sum evaluation of an oscillatory integral, and numerical evaluations of K_ℓ in Table 1. The paper also explicitly recalls in eq. (1) that the ℓ=1 palindromic-coefficient case has n deterministic zeros, giving E[N_n(0,2π)] = n + n/√3 + O(n^{3/4}).

Significance. If the main claim were valid, it would give an exact leading constant for a natural dependent-coefficient model and would quantify the increase over the classical i.i.d. case 2n/√3. The constant K_ℓ is defined as an integral with no fitted parameters, and the numerical table is reproducible from the formula, which are strengths. However, the theorem as stated is not supported: the ℓ=1 case is contradicted by the paper's own eq. (1), and the proof is carried out only under parity restrictions that exclude infinitely many n (and, for ℓ=2, all n). There is also an unsupported localization step involving F_ℓ. These are load-bearing issues, so the current manuscript should not be accepted.

major comments (3)

[Section 2, Theorem 2.1 and eq. (1)] As stated, Theorem 2.1 is false for ℓ=1. For ℓ=1 the coefficients are simply palindromic, and eq. (1) of the paper gives E[N_n(0,2π)] = n + n/√3 + O(n^{3/4}) because of n deterministic zeros. Since K_1=1/2, Theorem 2.1 instead predicts the leading term n/√3, omitting the deterministic term n. The remark after the theorem and the sentence after eq. (11) repeat this omission, and the Kac–Rice argument is formally inapplicable at the deterministic zeros, where A(x)=0; for ℓ=1 the excluded set F_ℓ only removes a neighborhood of x=0, not the n zeros of cos(nx/2). The theorem must either exclude ℓ=1 or state the correct ℓ=1 asymptotics.
[Proof of Theorem 2.1, parity reduction] The proof is explicitly restricted in two places: it assumes 'n−ℓ is odd' to obtain symmetry of A, B, C about x=π/2, and later says 'for simplicity we let n be even' in the Riemann-sum step. These two assumptions together cover only the subsequence with ℓ odd and n even. The theorem, however, claims the result for every fixed ℓ and every n in the stated range. For example, when ℓ=2, the condition n−ℓ odd forces n odd while the Riemann-sum step requires n even, so no n is covered; when ℓ is odd and n is odd, neither condition holds. No argument is supplied for the complementary parities, where the π/2 symmetry and the reduction of I_ℓ(n) to half intervals fail. The O(n^{2/3}) asymptotic is therefore established only on a parity-restricted subsequence, not for all n as claimed.
[Proof of Theorem 2.1, before eq. (11)] The step 'E[N_n(F_ℓ)] = O(n^{1-a}) by Lemma 3.1' is not justified. Lemma 3.1 concerns only the interval (0, n^{-a}), whereas F_ℓ also contains intervals centered at iπ/ℓ for i=1,...,[ℓ/2]. The behavior of V_n near those points is not equivalent to its behavior near 0, and a separate localization estimate would be needed. Since the subsequent Kac–Rice evaluation is performed only on E_ℓ, this gap affects the error term of the main asymptotic.

minor comments (3)

[Proof of Theorem 2.1, near eq. (7)] The text says 'sec(ℓx)=O(n^a)' where the subsequent division by sin(ℓx) indicates that csc(ℓx) is meant; please correct the notation.
[Proof of Theorem 2.1, symmetry statement] The statement that g(s,t) is 'symmetric about the line ⟨π/2,π/2,r⟩' is unclear; the needed symmetry of g under the relevant reflection in (s,t) should be stated explicitly.
[Lemmas 3.1 and 3.2] The proofs of Lemmas 3.1 and 3.2 are not included and are attributed to the unpublished preprint [23]; please either include proofs or give a published reference, since these lemmas are used in the main proof.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: K_ell is an explicit integral limit, not a fitted parameter; the self-cited lemmas are auxiliary estimates; the parity and ell=1 gaps are correctness issues, not circularity.

full rationale

The central claim is not circular. The constant K_ell is defined as a double integral built from the block structure via the Kac-Rice integrand, and the values in Table 1 are numerical evaluations of that integral, not fitted parameters. No fitted input is renamed as a prediction. The two lemmas taken from the author's own preprint [23] (Lemmas 3.1 and 3.2) are auxiliary estimates for endpoint contributions and trigonometric sums; they do not define K_ell or assume the ell-block asymptotic, so the leading-order prediction has independent mathematical content. The proof does have genuine limitations: it assumes 'n-l is odd' for the pi/2 symmetry and later 'for simplicity we let n be even,' so Theorem 2.1 is proved only along a parity-restricted subsequence, and equation (1) acknowledges n deterministic zeros at ell=1, which is inconsistent with the theorem's K_1=1/2 statement. These are correctness and completeness concerns, not cases where the conclusion is an input by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; Kℓ is an integral defined by the coefficient model. The proof rests on the generalized Kac-Rice formula, two trigonometric-sum lemmas imported from the author's earlier preprint [23], and the parity/symmetry reduction. No new physical or mathematical entities are postulated.

assumptions (3)

domain assumption Kac-Rice formula (∗) from [22], stated for iid coefficients, applies to Gaussian coefficients with non-trivial covariance and to the limiting exclusion of singular points.
The quoted proposition is stated for iid coefficients; the proof applies it to palindromic-block Gaussian coefficients and to Eℓ after removing neighborhoods of A=0. The generalized Kac-Rice theorem is standard, but the paper does not state conditions covering its use.
domain assumption Lemmas 3.1 and 3.2 of the author's earlier preprint [23] bound expected zeros near zero and trigonometric sums.
Both lemmas are imported without proof from the cited arXiv preprint by the same author; the current theorem depends on them.
domain assumption Palindromic block arrangement places the remainder block in the middle, so paired indices differ by r+1.
The theorem's pairing a_{ℓj+k}=a_{ℓ(2m-1-j)+r+1+k} is the interpretation used in the proof; it is consistent with Definition 2.3 but should be stated explicitly.

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Pith. "Pith review of Real zeros of random cosine polynomials with palindromic blocks of coefficients." pith.science (2026). https://pith.science/paper/YH54PMRE

@misc{pith2026190808154,
  author       = {Pith},
  title        = {Pith review of: Real zeros of random cosine polynomials with palindromic blocks of coefficients},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YH54PMRE}},
  note         = {Machine review of arXiv:1908.08154}
}

abstract

It is well known that a random cosine polynomial $ V_n(x) = \sum_ {j=0} ^{n} a_j \cos (j x) , \ x \in (0,2 \pi) $, with the coefficients being independent and identically distributed (i.i.d.) real-valued standard Gaussian random variables (asymptotically) has $ 2n / \sqrt{3} $ expected real roots. On the other hand, out of many ways to construct a dependent random polynomial, one is to force the coefficients to be palindromic. Hence, it makes sense to ask how many real zeros a random cosine polynomial (of degree $ n $) with identically and normally distributed coefficients possesses if the coefficients are sorted in palindromic blocks of a fixed length $ \ell. $ In this paper, we show that the asymptotics of the expected number of real roots of such a polynomial is $ \mathrm{K}_\ell \cdot 2n / \sqrt{3} $, where the constant $ \mathrm{K}_\ell $ (depending only on $ \ell $) is greater than 1, and can be explicitly represented by a double integral formula. That is to say, such polynomials have slightly more expected real zeros compared with the classical case with i.i.d. coefficients.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Angst, F

H. Angst, F. Dalmao and G. Poly, On the real zeros of random trigonometric polynomials with d ependent coeﬃcients , Proc. Amer. Math. Soc. 147 (2019), 205-214

work page 2019

[2] [2]

Bogomolny, O

, E. Bogomolny, O. Bohigas, P. Leboeuf, Quantum chaotic dynamics and random polynomials , J. Statist. Phys. 85 (1996), no. 5-6, 639-679

work page 1996

[3] [3]

Bloch and G

A. Bloch and G. P´ olya, On the roots of certain algebraic equations , Proc. London Math. Soc. 33 (1932), 102-114

work page 1932

[4] [4]

A. T. Bharucha-Reid and M. Sambandham, Random polynomials, Academic Press, Orlando, (1986)

work page 1986

[5] [5]

C. K. Chui, Concerning rates of convergence of Riemann sums , J. Approx. Theory 4 (1971), 279-287

work page 1971

[6] [6]

Das, The average number of real zeros of a random trigonometric po lynomial, Proc

M. Das, The average number of real zeros of a random trigonometric po lynomial, Proc. Cambridge Philos. Soc. 64 (1968), 721-729

work page 1968

[7] [7]

Dunnage, The number of real zeros of a random trigonometric polynomia l, Proc

J.E.A. Dunnage, The number of real zeros of a random trigonometric polynomia l, Proc. London Math. Soc. 16 (1966), 53-84

work page 1966

[8] [8]

Edelman and E

A. Edelman and E. Kostlan, How many zeros of a random polynomial are real? , Bull. Amer. Math. Soc. 32 (1995), 1-37

work page 1995