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arxiv: 2312.14883 · v3 · submitted 2023-12-22 · 🧮 math.PR

Roots of polynomials under repeated differentiation and repeated applications of fractional differential operators

Brian C. Hall , Ching-Wei Ho , Jonas Jalowy , Zakhar Kabluchko This is my paper

Pith reviewed 2026-05-24 05:00 UTC · model grok-4.3

classification 🧮 math.PR
keywords random polynomialsroot distributionsfractional differential operatorslimiting measurestransport mapslogarithmic potentialfree probability
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The pith

The limiting root distribution after repeated applications of a fractional differential operator is the push-forward of the initial distribution under a transport map from PDE characteristics.

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

The paper starts from random polynomials of degree N with independent coefficients that converge in root distribution to some measure as N grows. It applies an operator of the form z^a (d/dz)^b a total of roughly Nt times and derives the new limiting root measure μ_t. This measure equals the image of the initial limiting measure under an explicit transport map T_t. The map is constructed by solving a PDE for the logarithmic potential of μ_t and integrating its characteristic curves. The construction recovers radial inward motion at constant speed for pure repeated differentiation and supplies a free-probability multiplication rule for general parameters.

Core claim

Starting from a random polynomial P^N of degree N with independent coefficients that possesses a limiting root distribution ν, the polynomial obtained after roughly Nt applications of z^a (d/dz)^b has limiting root distribution μ_t equal to the push-forward of ν under the map T_t, where T_t is the flow along the characteristic curves of the PDE satisfied by the logarithmic potential of μ_t.

What carries the argument

The transport map T_t obtained by flowing along the characteristic curves of the PDE satisfied by the logarithmic potential of the evolving measure.

If this is right

  • For repeated differentiation the roots move radially inward at constant speed until they reach the origin and disappear.
  • The transport map admits an interpretation in free probability as multiplication of an R-diagonal operator by an R-diagonal transport operator.
  • The construction supplies a push-forward characterization of the free self-convolution semigroup of radial measures on the complex plane.
  • When the operator involves integration the root dynamics become more complicated than simple transport.

Where Pith is reading between the lines

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

  • The same characteristic-flow method could be applied to other families of linear operators on polynomials whose action on the log-potential yields a closed PDE.
  • The free-probability multiplication rule may connect the root evolution to the multiplicative free convolution of circular elements.
  • Numerical checks on finite-N polynomials with Gaussian coefficients could confirm the predicted speed of radial motion before the large-N limit is taken.

Load-bearing premise

The initial random polynomial of degree N with independent coefficients possesses a limiting root distribution as N tends to infinity.

What would settle it

For a concrete initial distribution such as the circular law, compute the empirical roots of large-N polynomials after exactly floor(Nt) applications of the operator and test whether their empirical measure converges to the predicted push-forward.

Figures

Figures reproduced from arXiv: 2312.14883 by Brian C. Hall, Ching-Wei Ho, Jonas Jalowy, Zakhar Kabluchko.

Figure 1
Figure 1. Figure 1: The smaller roots (left) travel radially at constant speed until they hit the origin and die before time t. The larger roots (right) travel radially at constant speed without hitting the origin. The blue dots show the roots of all the polynomials with time s < t, while the red dots show the roots with time t. Shown for t = 0.4 starting from a Weyl polynomial with N = 300. transform of µt, as in (1.2). Sinc… view at source ↗
Figure 2
Figure 2. Figure 2: The degree-increasing case with a = 5/2 and b = 3/2. The roots move radially inward without reaching the origin. Shown for N = 150 and t = 1/5, starting from a Weyl polynomial. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● … view at source ↗
Figure 3
Figure 3. Figure 3: The repeated integration case (a = 0, b = −1) starting from a Weyl polynomial. The blue dots show the roots of all the polynomials with time s < t, while the red dots show the roots with time t. Shown for t = 0.15 and N = 300 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The roots of QN t (left) and RN t (right) with a = 0 and b = 0.999, starting from a Weyl polynomial. Shown with N = 1000 and t = 1/2. of the original polynomial, together with N t roots at the origin. By contrast, RN t has a positive fraction of its roots concentrated near a circle of some positive radius r. (The roots having magnitude greater than r are almost the same for the two polynomials.) Let us now… view at source ↗
Figure 5
Figure 5. Figure 5: The exponential profile (solid) and its concave majo￾rant (dashed) in the Weyl case, for t = 0.15 Thus, the limiting root distribution of QN t will be concentrated entirely on the ring of radius e −Mt = (1 + t) 1+t t t = (1 + t)  1 + 1 t t . When t is large, e −Mt ≈ (1 + t)e. 8.2. Singular behavior: the Weyl polynomial case. As our next example, we consider the Weyl polynomials, which correspond to g0(α)… view at source ↗
Figure 6
Figure 6. Figure 6: The roots of the original exponential polynomial, di￾lated by a factor of 1 + t (left), and the roots of P N t (right). The inner circle has radius t. Shown for t = 0.3 and N = 1, 000. the exponential polynomials is easily computed using Theorem 3.4 and µ0 assigns mass r to the disk of radius r, for all 0 ≤ r ≤ 1. In that case, Example 3.11 with a = 0 and b = −1 applies. The limiting root distribution µt o… view at source ↗
read the original abstract

We start with a random polynomial $P^{N}(z)$ of degree $N$ with independent coefficients. We then consider a new polynomial $P_{t}^{N}$ obtained by $\lceil Nt\rceil$ applications of a fractional differential operator of the form $z^{a} (d/dz)^{b},$ where $a$ and $b$ are real numbers. When $b>0,$ we compute the limiting root distribution $\mu_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $\mu_{t}$ is the push-forward of the limiting root distribution of $P^{N}$ under a transport map $T_{t}$. The map $T_{t}$ is defined by flowing along the characteristic curves of a PDE satisfied by the log potential of $\mu_{t}.$ In the special case of repeated differentiation, our results may be interpreted as saying that the roots evolve radially \textit{with constant speed} until they hit the origin, at which point, they cease to exist. For general $a$ and $b,$ the transport map $T_{t}$ has a free probability interpretation as multiplication of an $R$-diagonal operator by an $R$-diagonal \textquotedblleft transport operator.\textquotedblright As an application, we obtain a push-forward characterization of the free self-convolution semigroup $\oplus$ of radial measures on $\mathbb{C}$. We also consider the case $b<0,$ which includes the case of repeated integration. More complicated behavior of the roots can occur in this case.

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.

Referee Report

2 major / 1 minor

Summary. The paper starts with a random polynomial P^N(z) of degree N with independent coefficients and considers the polynomial obtained after applying the fractional differential operator z^a (d/dz)^b a total of ⌈Nt⌉ times. It claims to compute the limiting root distribution μ_t as N→∞, showing that μ_t is the push-forward of the initial limiting root distribution under a transport map T_t obtained by flowing along characteristics of a PDE satisfied by the log potential of μ_t. Special cases include repeated differentiation (roots move radially at constant speed until reaching the origin) and a free-probability interpretation as multiplication by an R-diagonal transport operator; an application yields a push-forward characterization of the free self-convolution semigroup of radial measures on ℂ. The case b<0 (including repeated integration) is also treated.

Significance. If the claims hold under appropriate conditions, the work supplies a dynamical description of root evolution under fractional operators that links random polynomials to free probability and gives an explicit characterization of the radial free self-convolution semigroup. The transport-map construction via PDE characteristics is a potentially useful technical contribution when the initial limiting measure exists.

major comments (2)
  1. [Abstract] Abstract and opening paragraphs: the central claim that μ_t is computed for general independent coefficients presupposes the existence of a limiting root distribution μ_0 for the initial P^N. No moment or distributional hypotheses are stated, yet existence of such limits is a non-trivial prerequisite (typically requiring e.g. finite log-moments or Gaussianity) that is not proved or even explicitly assumed in the manuscript; without it the push-forward construction cannot be applied.
  2. [Abstract] The derivation that the evolved measure is exactly the push-forward under the characteristic flow T_t of the log-potential PDE is presented as the main result, but the manuscript supplies no error estimates, tightness arguments, or verification that the limiting empirical measure converges to this transported measure; the soundness of the transport step therefore cannot be assessed from the given text.
minor comments (1)
  1. The notation for the fractional operator z^a (d/dz)^b should include an explicit definition or reference for non-integer exponents a and b.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the assumptions and rigor. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the central claim that μ_t is computed for general independent coefficients presupposes the existence of a limiting root distribution μ_0 for the initial P^N. No moment or distributional hypotheses are stated, yet existence of such limits is a non-trivial prerequisite (typically requiring e.g. finite log-moments or Gaussianity) that is not proved or even explicitly assumed in the manuscript; without it the push-forward construction cannot be applied.

    Authors: We agree that existence of the initial limiting measure μ_0 is a prerequisite not proved in the paper. The abstract and main text presuppose this limit exists when they refer to computing μ_t as its push-forward under T_t. The manuscript's focus is the evolution step assuming μ_0 is given, rather than proving existence (which is known in the literature under conditions such as i.i.d. Gaussian coefficients or finite log-moments). In revision we will explicitly state the assumption in the abstract and introduction, with a brief reference to standard sufficient conditions from the random-polynomial literature. revision: yes

  2. Referee: [Abstract] The derivation that the evolved measure is exactly the push-forward under the characteristic flow T_t of the log-potential PDE is presented as the main result, but the manuscript supplies no error estimates, tightness arguments, or verification that the limiting empirical measure converges to this transported measure; the soundness of the transport step therefore cannot be assessed from the given text.

    Authors: The derivation obtains the explicit form of μ_t by evolving the log-potential, deriving the associated PDE, and solving along characteristics to produce the transport map T_t; the limiting distribution is then defined as the push-forward of μ_0. We acknowledge that the manuscript contains no quantitative error bounds, tightness proofs, or direct verification that the empirical measures converge to this transported limit. The result is therefore formal in the sense that it describes the candidate limit under the standing assumption that the initial empirical measures converge to μ_0 and that the transport applies in the limit. In revision we will add a clarifying remark in the introduction and main theorem statement to this effect, and we will indicate the additional analytic work that would be needed for a fully rigorous convergence proof. revision: yes

Circularity Check

0 steps flagged

No circularity; forward evolution from assumed initial limit

full rationale

The paper assumes existence of the initial limiting root measure μ_0 for P^N with independent coefficients (a standard external prerequisite in random polynomial theory) and derives the evolved measure μ_t as its push-forward under the characteristic flow of the log-potential PDE. No equation reduces the transport map T_t or μ_t to a fitted quantity or self-definition; the construction is explicit from the PDE and free-probability interpretation. No self-citations are invoked as load-bearing uniqueness results. The derivation chain is therefore self-contained once the initial convergence (independent of the present work) is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a limiting root distribution for the initial polynomial and on the well-posedness of the characteristic flow for the log-potential PDE; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • domain assumption A random polynomial of degree N with independent coefficients possesses a limiting empirical root measure as N tends to infinity.
    The paper begins with this measure and evolves it; the assumption is invoked at the outset of the abstract.
  • domain assumption The fractional differential operator z^a (d/dz)^b generates a well-defined evolution of the root measure via the stated PDE.
    The transport map is defined by flowing along characteristics of that PDE.

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