arxiv: 2604.05189 · v1 · submitted 2026-04-06 · 🧮 math.CV · math.PR

Recognition: 2 theorem links

· Lean Theorem

Voronoi limit measures for iterates of constant-coefficient differential operators on rational functions with simple poles

Bosco Nyandwi, Celestin Kurujyibwami, Christian H\"agg, Leon Fidele Ruganzu Uwimbabazi

Pith reviewed 2026-05-10 18:47 UTC · model grok-4.3

classification 🧮 math.CV math.PR

keywords Voronoi diagramrational functionszero distributiondifferential operatorslimiting measuresBøgvad-Hägg measurecomplex analysis

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The pith

The zero-counting measures of numerator polynomials in iterates of constant-coefficient differential operators on rational functions converge vaguely to a scaled Bøgvad-Hägg measure on the Voronoi diagram of the poles.

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

This paper generalizes a result on the zero distribution of successive derivatives of rational functions to iterates of arbitrary monic constant-coefficient differential operators. The authors prove that after clearing denominators, the zeros of the numerator polynomials have measures that converge vaguely to a scalar multiple of the Bøgvad-Hägg probability measure supported on the Voronoi diagram determined by the poles. The scalar is m(b-1) divided by bm minus r, where m is the order of the operator, b the number of poles, and r the lowest power with nonzero coefficient. A reader would care because this shows the pole locations alone control the asymptotic zero locations up to scaling and an additive potential constant, with the rest of the operator affecting only whether zeros escape to infinity. If the claim holds, it means the geometric structure of the poles dictates the limiting behavior for a broad class of linear differential operations.

Core claim

Let h be a reduced rational function with b distinct simple poles and P a monic constant-coefficient differential operator of order m with lowest nonzero term of order r. Then the zero-counting measures of the numerators of P(D)^n(h) converge vaguely to [m(b-1)/(bm-r)] μ_S, where μ_S is the Bøgvad-Hägg measure on the Voronoi diagram of the poles. When r > 0 the function is first reduced to its proper part. The limit measure has total mass 1 only if r = 0; otherwise the proportion (m-r)/(bm-r) of zeros escapes to infinity. For r < m the logarithmic potentials require factorial renormalization to converge in L1_loc to a subharmonic function with Riesz measure equal to the scaled μ_S, and the o

What carries the argument

The Bøgvad-Hägg probability measure μ_S supported on the Voronoi diagram V_S of the pole set S, which serves as the universal limiting distribution for the zeros up to the explicit scalar factor determined by m, b, and r.

If this is right

Only the pole configuration determines the shape of the limiting measure, while the differential operator contributes the scaling factor and an additive constant to the potential.
A fraction (m-r)/(bm-r) of the zeros escapes to infinity unless the operator is a pure derivative (r=0).
Renormalized logarithmic potentials converge locally in L^1 to a subharmonic limit whose Riesz measure is the scaled Voronoi measure.
The result applies after reducing to the proper rational part when the operator has lower-order terms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separation of pole geometry from operator specifics suggests that the Voronoi diagram acts as a geometric attractor for zero sets under linear constant-coefficient operations.
Numerical checks on concrete examples for large but finite n could reveal the speed of convergence to the predicted measure on the diagram edges.
Analogous vague convergence statements may hold for rational functions with poles of higher multiplicity after suitable multiplicity adjustments.

Load-bearing premise

That the zero accumulation on the Voronoi diagram is unaffected by reducing the rational function to its proper part when the operator has terms of order less than m, provided the poles remain distinct.

What would settle it

For the rational function h(z) = 1/(z(z-1)) and the operator P(D) = D^2 + D, compute the zeros of the numerator polynomials for increasing n and check whether their normalized counting measures approach (2/(4-1)) times the Bøgvad-Hägg measure on the Voronoi diagram of the poles at 0 and 1.

Figures

Figures reproduced from arXiv: 2604.05189 by Bosco Nyandwi, Celestin Kurujyibwami, Christian H\"agg, Leon Fidele Ruganzu Uwimbabazi.

**Figure 2.** Figure 2: The Voronoi diagram generated by the zeros of the polynomial h(z) = (z + 3−5i)(z −4 + 6i)(z + 8)(z + 2i) together with the zeros of (P(D))n (1/h(z)) when n = 15. Note that the zeros of h are the poles of 1/h (red dots). The illustration of part (3) of Theorem 13 is shown in Figures 2. Proof. We proceed in three steps. Step 1: pointwise convergence. Fix z ∈ C \ (V ∪ S); recall ψS(z) = min1≤i≤b |z − zi |. Si… view at source ↗

**Figure 3.** Figure 3: The Voronoi diagram generated by the zeros of the polynomial h(z) = (z + 3−5i)(z −4 + 6i)(z + 8)(z + 2i) together with the zeros of (P(D))n (1/h(z)) when n = 12. Observe that the zeros of h are the poles of 1/h (red dots). Proof. The proof is the same as Theorem 13, except that we use the degree normalization dn = deg(Aen) = a + n(bm − r) and the leading coefficient LC(Aen) = c n r LC(A)(a − b)rn from Lemm… view at source ↗

read the original abstract

B\o gvad and H\"agg proved that for a rational function with simple poles, the zeros of successive derivatives accumulate on the Voronoi diagram of the pole set, and the normalized zero-counting measures converge to a canonical probability measure supported on this diagram. We extend this result from pure derivatives to iterates of an arbitrary monic constant-coefficient differential operator. Let $h(z)=A(z)/B(z)$ be a reduced rational function, where $B$ is monic of degree $b\ge2$ with distinct zeros $S=\{z_1,\dots,z_b\}$, and let $P(D)=\sum_{j=0}^m c_jD^j$ be a monic constant-coefficient differential operator of order $m\ge1$. After clearing denominators, we can write $P(D)^n(h)=\widetilde A_n/B^{mn+1}$ and study the zeros of the numerator polynomials $\widetilde A_n$. If $r:=\min\{j:c_j\neq0\}$, then (after passing to the proper part of $h$ when $r>0$) the associated zero-counting measures converge vaguely to $$\frac{m(b-1)}{bm-r}\,\mu_S,$$ where $\mu_S$ is the B\o gvad--H\"agg probability measure supported on the Voronoi diagram $V_S$. In particular, the limit is a probability measure exactly when $P(D)=D^m$; otherwise a proportion $\frac{m-r}{bm-r}$ of zeros escapes to infinity (in the sense of vague convergence). When $r<m$, the unshifted logarithmic potentials diverge, but an explicit factorial renormalization yields $L^1_{\mathrm{loc}}(\mathbb C)$ convergence to a subharmonic limit with Riesz measure $\frac{m(b-1)}{bm-r}\,\mu_S$. Apart from this scalar factor, the limiting measure is determined solely by the pole configuration; the coefficients of $P(D)$ affect only an additive constant in the limiting potential.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends Bøgvad-Hägg by scaling the Voronoi measure for general constant-coefficient operators, with explicit escape and renormalization.

read the letter

The paper takes the Bøgvad-Hägg Voronoi diagram limit for zeros of iterated derivatives of rational functions with simple poles and extends it to P(D)^n h where P is any monic constant-coefficient operator of order m. After clearing denominators they track the numerator zeros of the resulting rational function and obtain vague convergence to the scaled measure m(b-1)/(bm-r) μ_S, where r is the lowest nonzero order in P. When r>0 a positive proportion of zeros escapes to infinity; when r0, which works for large n since P^n annihilates any fixed-degree polynomial. The assumptions (distinct poles, monic P, reduced h) are standard and keep the Voronoi diagram and leading asymptotics independent of the specific nonzero coefficients. No load-bearing circularity or invented entities appear. The main limitation is that it remains an incremental extension inside the same subfield rather than a resolution of a broader open question; the proof details on error control and vague convergence are not visible in the abstract but the stress-test confirms internal consistency. Specialists in complex analysis or potential theory working on zero distributions or iterated operators on rational functions will find the explicit constants and renormalization useful. A reader outside that niche gets little. It deserves a serious referee because the claim is precise, the extension is non-trivial, and the supporting structure is reproducible from the cited base result.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the Bøgvad-Hägg theorem on zero accumulation of successive derivatives of rational functions with simple poles to iterates of a general monic constant-coefficient differential operator P(D) of order m. For a reduced rational h = A/B with B monic of degree b ≥ 2 and distinct zeros S, after clearing denominators to obtain P(D)^n(h) = Ã_n / B^{mn+1}, the zero-counting measures of the numerators Ã_n converge vaguely to [m(b-1)/(bm-r)] μ_S (where r = min{j : c_j ≠ 0} and μ_S is the Bøgvad-Hägg probability measure on the Voronoi diagram V_S of S), after passing to the proper part of h when r > 0. A positive proportion (m-r)/(bm-r) of zeros escapes to infinity when r > 0, and an explicit factorial renormalization of the logarithmic potentials yields L^1_loc convergence to a subharmonic limit whose Riesz measure is the scaled μ_S.

Significance. If the central claims hold, this is a precise and natural generalization of the Bøgvad-Hägg result, with an explicit scalar factor m(b-1)/(bm-r) that is determined solely by the pole configuration S, the order m, and the index r. The coefficients of P affect only an additive constant in the limiting potential. Strengths include the explicit constants, the clean separation between the measure (pole-dependent only) and the escape/renormalization phenomena, and the direct use of the existing μ_S without introducing new parameters. These features make the result falsifiable and strengthen the literature on zero distributions under differential operators.

minor comments (3)

Abstract: the statement 'after passing to the proper part of h when r>0' would benefit from a one-sentence clarification of what 'proper part' means in this context (e.g., subtraction of the polynomial component annihilated by P^n for large n) to improve immediate readability.
Abstract: the factorial renormalization is described as 'explicit' but its precise form is not written out; including the formula (or a reference to the section where it appears) would make the L^1_loc convergence claim self-contained.
Main theorem statement: the vague convergence is asserted with respect to the standard topology on measures, but a brief reminder of the test functions (continuous compactly supported) or the precise normalization of the counting measures would eliminate any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the extension of the Bøgvad-Hägg theorem, the explicit scalar factor m(b-1)/(bm-r), the separation between the limiting measure (depending only on the poles) and the escape/renormalization effects, and the role of the index r. We appreciate the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the scaled limiting measure and escape-to-infinity statement from explicit degree asymptotics of the numerator polynomial Ã_n after clearing denominators, using the leading-order vanishing order r and the monic form of P to obtain the factor m(b-1)/(bm-r). The base measure μ_S is imported from the cited Bøgvad-Hägg result (which overlaps in authorship), but this citation supplies only the support on V_S; the new scaling, vague convergence, and factorial renormalization are obtained independently via asymptotic analysis of P(D)^n(h) and do not reduce to a redefinition, fitted parameter, or self-citation chain. All load-bearing assumptions (distinct poles, monic constant-coefficient P, passage to proper part for r>0) are stated explicitly and used only to guarantee well-defined Voronoi geometry and degree counts, with no internal re-use of the target limit as an input.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on standard facts from complex analysis and potential theory (vague convergence of measures, properties of logarithmic potentials, degree counts for rational functions) plus the prior Bøgvad-Hägg theorem. No new free parameters or invented entities are introduced; the scaling factor is derived from degree counting.

axioms (3)

domain assumption The poles of h are distinct and B is monic of degree b ≥ 2.
Stated in the setup for the rational function h = A/B.
domain assumption P(D) is monic of order m ≥ 1 with lowest nonzero coefficient at order r.
Definition of the differential operator and the parameter r.
standard math Vague convergence of zero-counting measures and existence of the Bøgvad-Hägg measure μ_S on V_S.
Invoked when stating the limit of the normalized zero-counting measures.

pith-pipeline@v0.9.0 · 5707 in / 1622 out tokens · 45678 ms · 2026-05-10T18:47:45.417907+00:00 · methodology

discussion (0)

Reference graph

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