Zeros of repeated derivatives of random polynomials

Dong Yao; Renjie Feng

REVIEW 2 major objections 2 minor 24 references

Zeros of repeated derivatives of random polynomials

T0 review · 2 major / 2 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Zeros of Kac polynomials that have been differentiated $N_n$ times have a limiting global distribution that depends only on the limiting ratio $N_n/n$, with a collapse-and-rescaling regime when the ratio tends to 1.

desk verdict The Kac phase diagram for repeated derivatives is new and likely correct, but the main results all rest on a general theorem whose proof is a sketch with an unproved lower bound. read the letter →

arxiv 1908.00730 v1 pith:TC3P2243 submitted 2019-08-02 math.PR

Renjie Feng , Dong Yao This is my paper · ORCID

classification math.PR MSC 30C1560F1060G57

keywords randompolynomialsKaczerosofrepeateddifferentiationphasetransitionrescalinglimitsLegendre-Fencheltransformelliptic

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

Repeatedly differentiating a random polynomial changes where its zeros live, and this paper shows that for the classical Kac polynomials the change is controlled entirely by the limiting ratio $N_n/n$ of the number of derivatives to the degree. If the ratio tends to $0$, the zeros still converge to the uniform measure on the unit circle. If it tends to $a\in(0,1)$, the limiting zero measure becomes a rotationally invariant measure with $\mu^K_a(\mathbb{D}_r)=ar/((1-a)(1-r))$ for $r<1-a$ and $1$ beyond that radius. If the ratio tends to $1$, the zeros collapse to the origin, and after rescaling by $n/D_n$ they have the density $1/(2\pi|z|)$ on the unit disk. This matters because it identifies the critical growth order at which unit-circle clustering fails and shows that the limiting shape changes abruptly rather than gradually.

What carries the argument

The load-bearing object is the modified coefficient-asymptotics theorem, stated as Theorem 2: for random polynomials whose coefficient magnitudes are approximated by a profile $p(k/L_n\wedge T_0)$, the normalized zero measure converges to a rotationally invariant measure determined by the Legendre-Fenchel transform of $-\log p$, namely $\mu(\mathbb{D}_r)=I'(\log r)$. The proof is sketched in Appendix A and adapts the upper- and lower-bound estimates of the cited theorem, with the lower-bound estimate (71) asserted to follow by replacing $n$ with $L_n$. For each case, the paper chooses $L_n$, a profile $p$, and a rescaling factor, then uses standard factorial asymptotics to show that the derivative coefficients $f_{k,n}=(k+N_n)!D_n!/(k!n!)$ fit the profile, so Theorem 2 delivers the limiting measure. The rescaling cases work by multiplying the coefficients by $R_n^{D_n-k}$ or $R_n^{-(D_n-k)/2}$ so that the dominant coefficient terms obey a new profile.

What would settle it

Simulate zeros of $K_n^{(N_n)}$ for $N_n=\lfloor n/2\rfloor$ with, say, i.i.d. standard complex Gaussian coefficients: the theorem predicts $(1/D_n)\mu^K_{D_n}(\mathbb{D}_r)\to ar/((1-a)(1-r))$ for $r<1-a$ and $1$ for $r\ge1-a$; a stable deviation from this radial curve at large $n$ would refute the central claim.

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Extended reading notes

Core claim

The paper establishes that for Kac polynomials $K_n(z)=\sum_{k=0}^n \xi_k z^k$ with i.i.d. nondegenerate coefficients satisfying $\mathbb{E}\log(1+|\xi_0|)<\infty$, the limiting global zero distribution of the $N_n$-th derivative is governed by $a=\lim N_n/n$. If $a=0$, then $(1/D_n)\mu^K_{D_n}$ converges in probability to the uniform measure on the unit circle. If $a\in(0,1)$, it converges to the rotationally invariant measure $\mu^K_a$ with $\mu^K_a(\mathbb{D}_r)=\frac{ar}{(1-a)(1-r)}$ for $0<r<1-a$ and $1$ otherwise. If $a=1$ and $D_n=n-N_n\to\infty$, the zeros collapse to the origin globally, while the rescaled polynomials $K_n^{(N_n)}(z/R_n)$ with $R_n=n/D_n$ have limiting zero density $\frac{1}{2\pi|z|}$ on the unit disk. If the residual degree $D_n=m$ is fixed, scaling by $n$ gives convergence in distribution to the random zeros of $\sum_{k=0}^m \xi_k z^k/k!$. For general random polynomials whose coefficients satisfy a profile assumption, the $a=0$ and $a\in(0,1)$ results carry over explicitly, while for $a=1$ the random elliptic polynomials give a different rescaling rate, $\sqrt{D_n/n}$, with density $r(\sqrt{4+r^2}-r)/2$, showing that the profile assumption alone does not determine the rescaling.

Load-bearing premise

The whole argument rests on the assumption that Theorem 2 is correct even though its proof is only sketched; in particular, the lower-bound estimate (71), asserted to follow from [9] by replacing $n$ with $L_n$, has to hold for every main limit theorem to go through.

Editorial extensions

If this is right

For Kac polynomials, the classical clustering of zeros near the unit circle survives repeated differentiation exactly when $N_n/n\to0$; any positive limiting ratio destroys it.
When $N_n/n\to1$ with $D_n\to\infty$, typical zeros of $K_n^{(N_n)}$ lie at distance of order $D_n/n$ from the origin, and after that rescaling the limiting density is $1/(2\pi|z|)$ on the unit disk.
For fixed residual degree $m$, the scaled zeros have a random limit described by the zeros of $\sum_{k=0}^m \xi_k z^k/k!$, rather than a deterministic measure.
For general random polynomials satisfying the coefficient-profile assumption, the $N_n/n\to0$ and $N_n/n\to a\in(0,1)$ limits are explicit in terms of the original profile, while the $N_n/n\to1$ rescaling is not determined by the profile alone, as the Kac and elliptic cases give different rates ($D_n/n$ versus $\sqrt{D_n/n}$).

Reading between the lines

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

Beyond the paper, the two rates $D_n/n$ and $\sqrt{D_n/n}$ suggest a general principle: the rescaling rate is set by the slowest coefficient growth near the top of the degree, so the quantities $\eta_n$ and $b_n$ introduced in the paper could predict when zeros collapse and at what speed.
One could conjecture a one-parameter family of limiting densities that interpolates between the $a\in(0,1)$ measure and the $1/(2\pi|z|)$ disk density as $N_n/n\to1$, with the two-step zooming heuristic in the paper indicating how the interpolation should look.
The fixed-degree limits for Kac and elliptic polynomials provide concrete test beds: perturbing the coefficient profile near the top of the degree while keeping the profile assumption intact should change the random limit in a way that tracks these baseline polynomials.

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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

Here's my read. The paper asks the right question—what happens to zeros of random polynomials when the number of derivatives grows with the degree—and gives a clean answer for Kac polynomials. Previous work only handled fixed-order derivatives. The phase diagram depending on lim N_n/n, with explicit limiting measures and rescaling limits, is genuinely new. The Kac coefficient estimates via Stirling are careful, the counterexample showing A1 is insufficient in the N_n/n→1 regime is honest, and the elliptic polynomial computation is concrete. This is a solid paper in intention and execution, modulo one gap.

The gap is Theorem 2. The appendix explicitly labels the proof a sketch. The upper bound (70) is derived, but the lower bound (71) is asserted: the set J from [9] is said to have at least |J|/2 L_n points after intersection with [0,(T0−δ_n)L_n], and the rest follows by replacing n with L_n. That is not a routine substitution. A2 lacks the R0 lower-bound condition from [9]; the index cutoff depends on δ_n; and the small-ball step needs rechecking with L_n in place of n. No details are supplied. Since Theorems 3, 5, and 6(2) all route through Theorem 2, the phase-transition conclusions are not fully supported as written. I suspect the gap is fixable—the heuristics look right—but it is a real gap.

Minor issue: the Legendre-Fenchel transform printed in Section 2.2 is wrong as written—it does not differentiate to the measure claimed just below. The measure is correct, so this is a typo, but it should be fixed.

Bottom line: this deserves a serious referee. The main results are important and likely true; the proof of Theorem 2 needs to be completed. I'd send it to peer review, with a request that the referee ask for a full proof of (71). Anyone working on zero distributions of random polynomials will want to read this, even in its current conditional form.

Referee Report

2 major / 2 minor

Summary. The paper studies the empirical measure of complex zeros of the N_n-th derivative of random polynomials, primarily Kac polynomials with i.i.d. coefficients. The main result for Kac polynomials (Theorem 3) is that the limiting global distribution depends on a = lim N_n/n: for a=0 the zeros uniformly cluster on the unit circle; for a in (0,1) the limit is an explicit rotationally invariant measure (Eq. (15)); for a=1 the zeros collapse to the origin, and after rescaling by n/D_n one obtains a measure with density 1/(2π|z|) on the unit disk (Eq. (17)-(18)). The fixed-degree case is treated by Rouché's theorem (Theorem 4), yielding a random limit related to f_m(z)=Σ ξ_k z^k/k!. The results are extended to general random polynomials satisfying the Kabluchko-Zaporozhets assumptions (Theorem 5), and for elliptic polynomials a different rescaling rate √(n/D_n) with an explicit limiting measure is computed (Theorem 6). The proofs route through a modified Kabluchko-Zaporozhets theorem (Theorem 2), whose proof is sketched in Appendix A.

Significance. If the results are fully established, this is a substantial contribution to the theory of random polynomials. It answers a natural question about the effect of dependent-on-degree differentiation and identifies a sharp phase transition controlled by the ratio N_n/n. The limiting measures are explicit and parameter-free, and the elliptic example demonstrates that the general A1 assumptions are insufficient to determine the rescaling, which is an honest and valuable observation. The coefficient asymptotics in Sections 2 and 4 are computed carefully and in detail, and the two-step heuristic connecting the fixed-degree and growing-degree regimes is illuminating. The main weakness is that the paper's central tool, Theorem 2, is only sketched: the critical lower-bound estimate (71) is asserted to follow from the Kabluchko-Zaporozhets proof by a substitution, but no complete argument is supplied. Since Theorems 3, 5, and 6 all inherit their limits from Theorem 2, the main conclusions are conditional on a rigorous proof of that theorem.

major comments (2)

[Appendix A, Eq. (71)] The proof of Theorem 2 is the load-bearing step for all main limit theorems, but the lower-bound estimate (71) is not proved. The text states that the set J from the proof in [9] has at least |J|/2 L_n points after intersection with [0,(T0-δ_n)L_n] and that the rest follows by replacing n with L_n. This is not a routine substitution: the assumptions A2 do not include an analogue of condition A1.4 (the R0 lower-bound condition), and the index range itself depends on δ_n. No argument is given that the intersection preserves linear cardinality when the supremum of I is attained at T0 and δ_n > 0, nor that the small-ball/Paley-Zygmund step of [9] works under A2 with L_n in place of n. Since the limiting measures in Theorem 3(2)-(3), Theorem 5, and Theorem 6(2) are inherited from Theorem 2, an unproved (71) leaves the phase-transition conclusions unsupported. A complete proof of Theorem 2 under A2 is needed before the main results are fully verified.
[§2.2, after Eq. (43)] The displayed Legendre-Fenchel transform I(s) is incorrect as written. For the function log f1(t) actually used, with the corrected definition log f1(t) = a log(t+a) + t log(1+a/t) + (1-a)log(1-a), the transform for s < log(1-a) should contain a term -a s and the denominator e^{-s}-1, not e^{-s}-1+a. Differentiating the printed formula gives a/(1-r+ar) (up to sign) rather than ar/(1-r); the measure claimed in (15) is the derivative of the correct transform. The final formula (15) appears to be correct, but the printed I(s) is internally inconsistent and must be corrected.

minor comments (2)

[§2.2, Eq. (45)] In the line preceding Eq. (45), the expression "1/n f_{k,n}" should read "(1/n) log f_{k,n}"; the logarithm is missing from the displayed formula.
[Title and abstract] The title and abstract contain spacing/OCR artifacts such as "REPEA TED" and "POL YNOMIALS"; these should be cleaned up in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the limiting measures are derived from explicit coefficient asymptotics fed into an external theorem, not from fitted inputs or self-citations.

full rationale

The paper's derivation chain is not circular. The coefficients of the differentiated polynomials are computed explicitly (e.g., f_{k,n} in equation (30), the rescaled Kac coefficients in (52), and the elliptic coefficients in (65)), and the limiting zero measures are then obtained by checking the A2 assumptions and applying Theorem 2, whose proof is an adaptation of the external Kabluchko-Zaporozhets theorem [9]. No parameter is fitted to the target measures and then renamed as a prediction; the measures are outputs of Legendre-Fenchel transforms computed from the explicitly identified functions f, u_a, and u_E. There are no self-citations carrying the load: [9] is independent prior work, not by the present authors. The main weakness is a proof gap, not circularity: in Appendix A, the crucial lower-bound estimate (71) is asserted, and the text says 'the rest proof follows the one in [9] by replacing n by L_n' without supplying the details. That is an incompleteness or correctness risk, but it does not make the conclusion an input to the derivation. Similarly, the reader's observation about the printed Legendre-Fenchel transform in Section 2.2 suggests an internal calculus inconsistency, but that is an accuracy issue, not a circularity issue. Accordingly, no circular step can be exhibited, and the honest finding is score 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no new mathematical entities. Its results are expressed through known constructions: Legendre-Fenchel transforms, rotationally invariant measures, and random analytic functions. The only nonstandard input is Theorem 2, a modified version of the KZ theorem, which is not fully proved.

assumptions (4)

domain assumption The random variables xi_k are i.i.d., nondegenerate, with E log(1+|xi_0|) < infinity and P(xi_0=0)=0.
Stated in Section 1.1, equations (2) and (3). Used throughout to guarantee the Kac polynomial degree n almost surely and the KZ-type limit theorems.
ad hoc to paper The modified Kabluchko-Zaporozhets theorem (Theorem 2) is valid under assumptions A2.
Theorem 2 is introduced in this paper and used to derive the limiting zero measures in Theorems 3, 5, and 6. Its proof in Appendix A is only sketched, with the lower bound (71) asserted to follow from [9].
domain assumption The coefficient profile functions p, f, and u_a are continuous and positive on their domains, and the associated A2 conditions hold.
A2 is the main regularity condition; the paper verifies it case by case using Stirling estimates. If these uniform convergence conditions fail, the limiting measures would not follow.
standard math Stirling's formula is applicable uniformly over k, including boundary cases k=0 and k=D_n.
Used throughout Section 2 to compute the leading-order behavior of the factorial-type coefficients f_{k,n}. The paper treats k=0 separately to maintain consistency.

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Cite this review

Pith. "Pith review of Zeros of repeated derivatives of random polynomials." pith.science (2026). https://pith.science/paper/TC3P2243

@misc{pith2026190800730,
  author       = {Pith},
  title        = {Pith review of: Zeros of repeated derivatives of random polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TC3P2243}},
  note         = {Machine review of arXiv:1908.00730}
}

abstract

It has been shown that zeros of Kac polynomials $K_n(z)$ of degree $n$ cluster asymptotically near the unit circle as $n\to\infty$ under some assumptions. This property remains unchanged for the $l$-th derivative of the Kac polynomials $K^{(l)}_n(z)$ for any fixed order $l$. So it's natural to study the situation when the number of the derivatives we take depends on $n$, i.e., $l=N_n$. We will show that the limiting global behavior of zeros of $K_n^{(N_n)}(z)$ depends on the limit of the ratio $N_n/n$. In particular, we prove that when the limit of the ratio is strictly positive, the property of the uniform clustering around the unit circle fails; when the ratio is close to 1, the zeros have some rescaling phenomenon. Then we study such problem for random polynomials with more general coefficients. But things, especially the rescaling phenomenon, become very complicated for the general case when $N_n/n\to 1$, where we compute the case of the random elliptic polynomials to illustrate this.

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Reference graph

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Pith tools

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

[1] [9]

Hough , M

J. Hough , M. Krishnapur , Y. Peres and B. Virag , Zeros of Gaussian Analytic Functions and Determinantal Point Processes. University Lecture Series, 51. American Mathematical Society, Providence, RI, 2009. x+154 pp

work page 2009

[2] [1]

Bloom and B

T. Bloom and B. Shiffman , Zeros of random polynomials on C^m . Math. Res. Lett. 14 (2007), no. 3, 469-479

work page 2007

[3] [2]

Bleher , B

P. Bleher , B. Shiffman and S. Zelditch , Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142 (2000), no. 2, 351-395

work page 2000

[4] [3]

Carleman, \" Uber die Approximation analytischer Funktionen durch lineare Aggregate von

T. Carleman, \" Uber die Approximation analytischer Funktionen durch lineare Aggregate von

work page

[5] [4]

Farmer and R

D. Farmer and R. Rhoades , Differentiation evens out zero spacings. (English summary) Trans. Amer. Math. Soc. 357 (2005), no. 9, 3789-3811

work page 2005

[6] [5]

Farmer and M

D. Farmer and M. Yerrington , Crystallization of Random Trigonometric Polynomials. J. Stat. Phys. 123 (2006), no. 6, 1219-1230

work page 2006

[7] [6]

Feng , Zeros of derivatives of Gaussian random polynomials on plane domain, preprint

R. Feng , Zeros of derivatives of Gaussian random polynomials on plane domain, preprint

work page

[8] [7]

Hanin , Correlations and Pairing Between Zeros and Critical Points of Gaussian Random

B. Hanin , Correlations and Pairing Between Zeros and Critical Points of Gaussian Random

work page