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

TL;DR

This work analyzes how zeros of random polynomials with independent coefficients evolve under repeated differentiation and generalized fractional differential operators $z^{a}(d/dz)^{b}$ as the degree grows. The authors establish a bulk (mean-field) description: the limiting root distribution at time $t$ is a push-forward of the initial distribution via a transport map defined by the PDE's characteristic curves for the log potential, with explicit formulas for the transport in many regimes. They extend the radial-rotation framework to fractional flows, derive exponential-profile behavior for the coefficient distributions, and prove push-forward theorems that connect root evolution to free probability constructs, including additive and multiplicative (radial) convolutions via $R$-diagonal operators. The paper also links these dynamics to existing free-probability results (COR) and provides a PDE/Hamilton–Jacobi perspective that clarifies root motion and mass creation/annihilation, including the complex behavior when $b<0$. Overall, it unifies differential-flow root dynamics with transport maps and free-probability interpretations, yielding precise descriptions of root motion, die-off, and the resulting radial measures.

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 $μ_{t}$ of $P_{t}^{N}$ as $N\rightarrow\infty.$ We show that $μ_{t}$ 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 $μ_{t}.$ 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.

Figures (6)

• 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 (Example \ref{['Weyl.example']}) with $N=300.$
• 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 (Example \ref{['Weyl.example']}). The blue dots show the roots at 15 different times less than $t$ and the red dots show the roots at time $t$.
• Figure 3: The repeated integration case ($a=0,$$b=-1$) starting from a Weyl polynomial (Example \ref{['Weyl.example']}). 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 $N=300.$
• Figure 4: The roots of $Q_{t}^{N}$ (left) and $R_{t}^{N}$ (right) with $a=0$ and $b=0.999,$ starting from a Weyl polynomial. Shown with $N=1000$ and $t=1/2.$
• Figure 5: The exponential profile (solid) and its concave majorant (dashed) in the Weyl case, for $t=0.15$

Theorems & Definitions (63)

• Definition 1.1
• proof : Heuristic derivation of Idea \ref{['singleDeriv.idea']}
• Example 1.3
• Remark 1.5
• Theorem 1.6: Push-forward Theorem for Repeated Differentiation
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4