Asymptotic root distribution of polynomials under repeated polar differentiation

Daniel Perales; Zhiyuan Yang

REVIEW 2 major objections 5 minor 10 references

Asymptotic root distribution of polynomials under repeated polar differentiation

T0 review · 2 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Repeated polar differentiation of real-rooted polynomials has a universal asymptotic zero distribution, obtained by conjugating fractional free convolution with the Möbius map that sends the differentiation point to infinity.

desk verdict Polar free convolution powers are a real new object and the main theorems are right, but the proof of Theorem 3.9 misses the boundary case ts=1—easily patched. read the letter →

arxiv 2508.18575 v1 pith:Z6ANZZVM submitted 2025-08-26 math.PR math.COmath.OA

Daniel Perales , Zhiyuan Yang This is my paper · ORCID

classification math.PRmath.COmath.OA MSC 30C1546L54

keywords polarderivativerootdistributionfreefractionalconvolutionMöbiustransformreal-rootedpolynomialscommutationrelationPoissonS-rationalmeasure

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

Repeated polar derivatives of real-rooted polynomials have a universal asymptotic root distribution. The paper proves that if a sequence of polynomials has zero distributions converging to a measure $\mu$, then applying the polar derivative $D_a$ enough times, so that the formal degree shrinks by a factor $t$, drives the zero distribution to $F^a_t\mu$, a measure built by pulling the ordinary fractional free convolution power back along the Möbius map $z\mapsto 1/(z-a)$. This turns the known principle that differentiation acts like free convolution on root distributions into a statement about a whole family of differential operators, one for each point $a$ of the extended real line. A second result, a commutation relation between $F^a_s$ and $F^b_t$ for distinct centers, follows from the commutativity of polar derivatives about different points. The paper also records explicit transformations of free Poisson and $S$-rational measures, and identifies a distribution with density $1/(\pi(1+x^2))$ that is invariant under every $F^a_t$.

What carries the argument

The load-bearing mechanism is the conjugation identity $D_a=T^{-1}_*\partial T_*$ for the Möbius map $T(z)=1/(z-a)$, together with the previously established theorem, quoted as Theorem 2.24 in the paper, that repeated ordinary differentiation converges to $F_t\mu=\operatorname{Dil}_{1/t}\mu^{\boxplus t}$. The paper transfers this identity to measures by defining $F^a_t\mu=T^{-1}_*F_t(T_*\mu)$ and using the weak continuity of Möbius pushforwards to pass from polynomial limits to measure limits. The second engine is the commutativity $D_aD_b=D_bD_a$ of polar derivatives, which, applied twice through the limit theorem, yields the commutation relation between $F^a_s$ and $F^b_t$.

What would settle it

Compute the zero distributions of $D_0^{m|n}H_n[\lambda;\cdot]$ for the one-parameter hypergeometric polynomials with $n/m\to t$ and compare them with $\operatorname{Dil}_{1/t}\pi_{t\lambda-t+1}$; a mismatch beyond numerical error would disprove the main limit theorem. A second check is to compute both sides of $F^a_sF^b_t\mu=F^b_{s'}F^a_{t'}\mu$ for a measure with atoms at $a$ and $b$, where the proof assigns the atomic masses explicitly.

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Extended reading notes

Core claim

The paper's central claim is Theorem 3.10: if $p_j$ are real-rooted polynomials of formal degree $n_j$, their zero distributions converge weakly to a probability measure $\mu$ on $\hat{\mathbb{R}}$, and $m_j$ is chosen so that $n_j/m_j\to t$ with $D^{m_j|n_j}_a p_j\ne0$ and $t\mu(\{a\})\le1$, then the zero distributions of $D^{m_j|n_j}_a p_j$ converge weakly to $F^a_t\mu$, defined as $F^a_t\mu=T^{-1}_*(\operatorname{Dil}_{1/t}(T_*\mu)^{\boxplus t})$ with $T(z)=1/(z-a)$. This is the exact polar analogue of the known result for ordinary differentiation, obtained by conjugating that result by a Möbius transformation. The second main result is the commutation relation $F^a_sF^b_t\mu=F^b_{s'}F^a_{t'}\mu$ whenever $st=s't'$ and $s+s'=1+st$, which follows from the commutativity of $D_a$ and $D_b$. The paper further determines the atom structure of $F^a_t\mu$, proves order-preservation properties, and applies the framework to free Poisson and $S$-rational measures and to a heavy-tailed invariant measure.

Load-bearing premise

The whole construction inherits the prior theorem that repeated ordinary differentiation of real-rooted polynomials with convergent zero distributions yields dilated fractional free convolution powers; if that theorem has regularity conditions not satisfied by the sequences allowed here, the polar limit formula would fail with it.

Editorial extensions

If this is right

For $a=0$, the operation $F^0_t$ is connected to multiplicative free convolution, yielding the explicit formula $F^0_t(\pi_\lambda)=\operatorname{Dil}_{1/t}\pi_{t\lambda-t+1}$ for the free Poisson family.
The free Poisson distributions are carried by both the usual and the polar semigroups, with a simple parameter shift and dilation, and the larger class of $S$-rational measures transforms by explicit parameter changes.
The distribution with density $1/(\pi(1+x^2))$ is a fixed point of $F^a_t$ for every $a\in\hat{\mathbb{R}}$ and every $t\ge1$.
Polar fractional powers preserve stochastic order: if $\mu\ll\nu$ on $(a,\infty)$, then $F^a_t\mu\ll F^a_t\nu$, and a similar monotonicity holds as the center $a$ moves.
The commutation relation $F^a_sF^b_t=F^b_{s'}F^a_{t'}$ with $st=s't'$ and $s+s'=1+st$ gives a new symmetry between the polar semigroups at distinct points of the extended real line.

Reading between the lines

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

If the commutation relation extends to fractional indices for polar-infinitely-divisible measures, as suggested in Section 6, the maps $F^a_t$ would generate a larger two-parameter action on measures; this could be tested by checking whether the constructed semigroup $B^b_a(t)=F^{1+t}_bF^{1/(1+t)}_a$ acts as a group on polar-infinitely-divisible measures.
The existence of a distribution invariant under every $F^a_t$ hints at stationary states for polar differentiation; a natural extension is to search for other fixed points among heavy-tailed distributions and to ask whether they correspond to polynomial sequences that interlace under $D_a$ the way Appell sequences do under ordinary differentiation.
The connection between $F^0_t$ and multiplicative free convolution suggests a computational route: polar root limits for many coefficient distributions could be calculated through $S$-transforms, which would make the formulas of Section 8 testable numerically for finite but large degree.

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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

Two things to know. First, this is a solid new contribution: it defines polar free convolution powers F^a_t = T^{-1}_* F_t T_* and proves the natural analogue of the repeated-differentiation theorem for polar derivatives (Theorem 3.10), plus a commutation relation F^a_s F^b_t = F^b_{s'} F^a_{t'} (Theorem 5.1) that is genuinely different from Belinschi–Nica in index bookkeeping. Second, the proof has a small boundary gap that should be fixed before publication, but it is easy to patch and does not threaten the main results.

The paper does a lot right. The reduction of polar differentiation to ordinary differentiation via Möbius conjugation is clean, and the transfer of the known theorem for M(R) to M(ˆR) works when the exponent is finite. The commutation relation has two proofs: one via polynomial sequences and one via an operator model. The examples are real: Laguerre/Marchenko–Pastur, S-rational measures, and the Cauchy invariance under F^a_t, the last being a nice observation. The authors also give a notion of polar free infinite divisibility and construct Belinschi–Nica-type semigroups, with proofs that follow the established pattern.

Where are the soft spots? The stress-test note identifies a genuine issue: in the proof of Theorem 3.9, when ts=1 (i.e. t μ({∞})=1), the exponent (t-ts)/(1-ts) has denominator zero and Theorem 2.24 does not apply. The claimed limit δ_∞ is correct—the mass at ∞ already tends to 1, so a one-line argument closes the gap—but as written the theorem is not fully proved in exactly the case the statement allows. The same issue propagates to Theorem 3.10 via conjugation. This is a minor patch, not a structural flaw.

The paper also leans on Theorem 2.24, cited from [JKM25], for the base differentiation result; that is reasonable, though any hidden regularity conditions there would propagate. The PDE section (Section 7) is explicitly non-rigorous and is flagged as such by the authors; it is not load-bearing. There are some typos and minor notation inconsistencies, but nothing confusing enough to block reading.

Overall: the central claims hold up, the new objects are natural and well-motivated, and the examples are independently verifiable. I would send this to a referee. It belongs in a good probability/analysis journal.

Referee Report

2 major / 5 minor

Summary. The paper studies the asymptotic root distribution of real-rooted polynomials under repeated polar differentiation. It introduces a polar free convolution power F_t^a on probability measures on the extended real line, proves that repeated polar differentiation of polynomials converges to this operation (Theorem 3.10), establishes a commutation relation F_s^a F_t^b = F_{s'}^b F_{t'}^a with st = s't' and s + s' = 1 + st (Theorem 5.1), and develops polar free infinite divisibility together with Belinschi-Nica type semigroups. The main tool is the conjugation identity D_a = T^{-1}_* ∂ T_* for polar derivatives, combined with the known asymptotic theorem for ordinary repeated differentiation (Theorem 2.24). The paper also contains explicit examples, including Marchenko-Pastur, Cauchy, and S-rational measures.

Significance. If the main theorems are fully rigorous, the paper provides a natural Möbius-conjugated extension of free fractional convolution, a nontrivial commutation relation, and a coherent theory of polar free infinite divisibility. The algebraic identities (Proposition 2.9, Corollary 2.10), the operator model in §5.2, and the explicit invariance and cumulant computations in §8 are valuable contributions. The clean semigroup structure and the Cauchy/Marchenko-Pastur examples make the paper attractive to researchers in free probability and asymptotic root distributions. However, the proof of the central asymptotic theorem (Theorem 3.9/3.10) contains a gap that must be repaired before the stated level of generality is established.

major comments (2)

[3.3] The proof asserts that "Since μ[p_j]({∞}) = k_j/n_j and μ[p_j] → μ, we have k_j/n_j converges to s := μ({∞})". This is false when the atom at ∞ in the weak limit arises from finite roots tending to infinity rather than from formal roots at infinity. For example, p_j(z) = (z-j)^{n_j} satisfies μ[p_j] = δ_j → δ_∞ on ˆR, while k_j/n_j = 0 for all j. Consequently, the reduction to the residual polynomial q_j and the invocation of Theorem 2.24 are not justified, because μ[q_j] need not converge in M(R). Since Theorem 3.10 is proved via Theorem 3.9, this gap affects the first main theorem as stated for measures in M(ˆR). The theorem may be true (it is consistent with [JKM25]), but the proof as written does not establish it. I recommend either citing the corresponding differentiation result from [JKM25] for measures on the extended real line, or supplying a truncation/compactness argument that separates the escaping-roots component from the formal-at-∞ component.
[3.3] The boundary case ts = 1 is not covered. Definition 3.1 gives F_t(μ) = ts δ_∞ + (1-ts) F^{(t-ts)/(1-ts)}ν for ts ≤ 1, but when ts = 1 the exponent (t-ts)/(1-ts) is undefined because the denominator is zero. In the proof of Theorem 3.9, the argument invokes Theorem 2.24 with this exponent and with the ratio (n_j-k_j)/(m_j-k_j) diverging, so the cited theorem does not apply. For instance, s = 1/2 and t = 2 gives t μ({∞}) = 1, which is explicitly allowed by the statement. The claimed limit δ_∞ (or δ_a after conjugation) is correct, but a separate limiting argument is needed. The same boundary case affects Theorem 3.10 when t μ({a}) = 1.

minor comments (5)

[3.4] The text "lim_{j→∞} n_j/k_j = t" should read "lim_{j→∞} n_j/m_j = t".
[1.2] The statement requires s',t' > 1, but if s = 1 or t = 1 then one of s',t' equals 1; the theorem should state ≥ 1, or explicitly restrict to s,t > 1.
[5.1] The displayed formula for (F_{t'}^a μ)({b}) is typeset ambiguously; it should be ((F_{t'}^a μ)({b})) = [s(t μ({b}) - 1) + 1]/(1 + st - s), with appropriate parentheses.
[3.1] For clarity, explicitly define F_t(μ) = δ_∞ when ts = 1 (consistent with the case ts > 1), so that the formula is not undefined at this point.
[8] The displayed formula for the constant c contains a typo: "c =:" should be "c :=".

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: polar free convolution powers are explicitly defined by Möbius conjugation of the established free fractional convolution, and the main limit and commutation theorems are proved by transferring the external differentiation theorem, not by assuming the target.

full rationale

I walked the chain from Definition 3.3 (F^s_a μ = T^{-1}_* F^s T_* μ) through Theorem 3.10 to Theorem 5.1. F^s_a is introduced as a definition, and Theorem 3.10 verifies the polynomial limit equals this defined map by applying the exact conjugation D_a = T^{-1}_* ∂ T_* (Corollary 2.10) and then invoking the external Theorem 2.24; no fitted parameter is renamed as a prediction. Definition 3.1 extends F_t to the extended real line using the same algebra that appears in the proof, but Theorem 3.9 still proves the limit formula from Theorem 2.24, so this is a stipulated extension, not a circular input. The commutation relation in Theorem 5.1 is obtained by applying Theorem 3.10 twice to the same polynomial sequence and using the algebraic identity D_a D_b = D_b D_a; the parameters s', t' are computed from the limits, not imposed. Examples such as the free Poisson identity F^t_0(π_λ) = Dil_{1/t} π_{tλ−t+1} are independently checked by cumulants, so they are not consequences of the definition alone. The cited differentiation theorem (Theorem 2.24) is attributed to [HK23, AGVP23, AFPU24, JKM25]; although one of these citations includes an author of the present paper, the theorem is independently established in the literature and is not a uniqueness claim imported solely from the authors' prior work. I flag two non-circular caveats: the proof of Theorem 3.9 does not handle the boundary case ts = 1, where the exponent (t − ts)/(1 − ts) has denominator zero and Theorem 2.24 cannot be invoked directly; and Section 7 is explicitly non-rigorous. Both are correctness concerns, not instances of circularity, and neither makes the main derivation equivalent to its inputs. The score of 1 reflects the presence of author-overlapping citations and the mildly tautological flavour of the Definition 3.1 extension, not a demonstrated circular step.

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

No free parameters are fitted to data; all quantities are fixed variables (t, s, a, b) in the statements. The paper's main new object, F^a_t, is a defined operation rather than an invented physical entity. The axioms listed are the external theorems and definitional choices on which the central claims rest; none are hidden or unfalsifiable.

assumptions (3)

domain assumption Fractional free convolution powers F_t μ := Dil_{1/t} μ^{⊞t} exist and are unique for all μ ∈ M(R), t ≥ 1, and repeated ordinary differentiation converges to F_t μ (Theorem 2.24).
Invoked in the proof of Theorem 3.10; the entire polar derivative limit reduces to this theorem after Möbius conjugation. The paper does not prove it.
ad hoc to paper The extension of F_t to measures with an atom at infinity follows Definition 3.1: for μ = s δ_∞ + (1-s)ν, F_t(μ) = ts δ_∞ + (1-ts) F^{(t-ts)/(1-ts)}ν for ts ≤ 1 and δ_∞ otherwise.
This definition is used to state Definition 3.3 of F^a_t and Theorem 3.10 for measures on ^R. It is natural but is a definitional choice, not forced by prior results.
standard math Weak continuity of Möbius pushforwards on M(^R) (Remark 2.21).
Used to pass the weak limit through T^{-1}_* in the proof of Theorem 3.10.

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Pith. "Pith review of Asymptotic root distribution of polynomials under repeated polar differentiation." pith.science (2026). https://pith.science/paper/Z6ANZZVM

@misc{pith2026250818575,
  author       = {Pith},
  title        = {Pith review of: Asymptotic root distribution of polynomials under repeated polar differentiation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z6ANZZVM}},
  note         = {Machine review of arXiv:2508.18575}
}

abstract

Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along M\"obius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions.

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Reference graph

Works this paper leans on

10 extracted references · 2 canonical work pages

[1]

Critical points of random polynomials and finite free cumulants.arXiv preprint arXiv:2506.08910 ,

[ACF25] Octavio Arizmendi, Andrew Campbell, and Katsunori Fujie. Critical points of random polynomials and finite free cumulants.arXiv preprint arXiv:2506.08910 ,

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The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation.International Mathematics Research Notices, 2024(13):10189–10218,

[COR24b] Andrew Campbell, Sean O’Rourke, and David Renfrew. The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation.International Mathematics Research Notices, 2024(13):10189–10218,

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Polynomials, roots, and interlacing

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv preprint math/0612833 ,

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Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators.Preprint arXiv:2312.14883,

[HHJK23] Brian Hall, Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators.Preprint arXiv:2312.14883,

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Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation

ROOT DISTRIBUTION UNDER REPEATED POLAR DIFFERENTIATION 33 [JKM25] Jonas Jalowy, Zakhar Kabluchko, and Alexander Marynych. Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation. Preprint arXiv:2504.11593,

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A semicircle law for derivatives of random polynomials

[HS22] Jeremy G Hoskins and Stefan Steinerberger. A semicircle law for derivatives of random polynomials. International Mathematics Research Notices , 2022(13):9784–9809,

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[2024]

Universality for roots of derivatives of entire functions via finite free probability.arXiv preprint arXiv:2410.06403 ,

[COR24a] Andrew Campbell, Sean O’Rourke, and David Renfrew. Universality for roots of derivatives of entire functions via finite free probability.arXiv preprint arXiv:2410.06403 ,

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S-transform in finite free probability

[AFPU24] Octavio Arizmendi, Katsunori Fujie, Daniel Perales, and Yuki Ueda. S-transform in finite free probability. Preprint arXiv:2408.09337,

arXiv

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Critical points of random polynomials and finite free cumulants.arXiv preprint arXiv:2506.08910 ,

[ACF25] Octavio Arizmendi, Andrew Campbell, and Katsunori Fujie. Critical points of random polynomials and finite free cumulants.arXiv preprint arXiv:2506.08910 ,

arXiv

[2] [5]

The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation.International Mathematics Research Notices, 2024(13):10189–10218,

[COR24b] Andrew Campbell, Sean O’Rourke, and David Renfrew. The fractional free convolution of R-diagonal elements and random polynomials under repeated differentiation.International Mathematics Research Notices, 2024(13):10189–10218,

work page 2024

[3] [6]

Polynomials, roots, and interlacing

[Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv preprint math/0612833 ,

arXiv

[4] [10]

Repeated differentiation and free unitary Poisson process

[Kab21] Zakhar Kabluchko. Repeated differentiation and free unitary Poisson process. Preprint arXiv:2112.14729,

arXiv

[5] [1995]

Free infinite divisibility, fractional convolution powers, and Appell polynomials

[Cam24] Andrew Campbell. Free infinite divisibility, fractional convolution powers, and Appell polynomials. Preprint arXiv:2412.20488,

arXiv

[6] [2013]

Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators.Preprint arXiv:2312.14883,

[HHJK23] Brian Hall, Ching-Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators.Preprint arXiv:2312.14883,

arXiv

[7] [2022]

Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation

ROOT DISTRIBUTION UNDER REPEATED POLAR DIFFERENTIATION 33 [JKM25] Jonas Jalowy, Zakhar Kabluchko, and Alexander Marynych. Zeros and exponential profiles of polynomials I: Limit distributions, finite free convolutions and repeated differentiation. Preprint arXiv:2504.11593,

arXiv

[8] [2023]

A semicircle law for derivatives of random polynomials

[HS22] Jeremy G Hoskins and Stefan Steinerberger. A semicircle law for derivatives of random polynomials. International Mathematics Research Notices , 2022(13):9784–9809,

work page 2022