[Paper Review] Conservation Laws for the Density of Roots of Polynomials under Differentiation

Stefan Steinerberger|arXiv (Cornell University)|Jan 27, 2020

Mathematical functions and polynomials27 references4 citations

TL;DR

This paper derives an infinite set of conservation laws governing the evolution of the root density $ u(t,x) $ of the $ (t imes n) $-th derivative of a degree-$ n $ polynomial with real, distinct roots as $ n \to \infty $. The key result shows that moments and correlation integrals of the root density evolve deterministically with $ (1-t)^k $, suggesting a nonlocal evolution involving the Hilbert transform.

ABSTRACT

Let $p_n(x)$ be a polynomial of degree $n$ having $n$ distinct, real roots distributed according to a nice probability distribution $u(0,x)dx$ on $\mathbb{R}$. One natural problem is to understand the density $u(t,x)$ of the roots of the $(t\cdot n)-$th derivative of $p_n$ where $0 < t < 1$ as $n ightarrow \infty$. We derive an extit{infinite} number of conversation laws for the evolution of $u(t,x)$. The first three are \begin{align*} \int_{\mathbb{R}}{ u(t,x) ~ dx} = 1-t, \qquad \qquad \int_{\mathbb{R}}{ u(t,x) x ~ dx} = \left(1-t ight)\int_{\mathbb{R}}{ u(0,x) x~ dx}, \qquad \int_{\mathbb{R}} \int_{\mathbb{R}} u(t,x) (x-y)^2 u(t,y) ~ dx dy = (1-t)^3 \int_{\mathbb{R}} \int_{\mathbb{R}} u(0,x) (x-y)^2 u(0,y) ~ dx dy. \end{align*} The author suggested that $u(t,x)$ might evolve according to a nonlocal evolution equation involving the Hilbert transform; this has been verified for two special closed form solutions -- these conservation laws thus point to interesting identities for the Hilbert transform. We discuss many open problems.

Motivation & Objective

• To understand the asymptotic distribution of roots of the $ (t \cdot n) $-th derivative of a high-degree polynomial with real, distinct roots as $ n \to \infty $.
• To identify conserved quantities that govern the time evolution of the root density $ u(t,x) $, where $ t \in (0,1) $.
• To explore whether the evolution of $ u(t,x) $ satisfies a nonlocal partial differential equation involving the Hilbert transform.
• To establish quantitative identities for moments and correlation integrals of the root density under differentiation.

Proposed method

• Derives conservation laws for the root density $ u(t,x) $ by analyzing the asymptotic behavior of the $ (t \cdot n) $-th derivative of a polynomial of degree $ n $ as $ n \to \infty $.
• Uses the initial root distribution $ u(0,x) $ as a probability measure on $ \mathbb{R} $, assuming it is sufficiently smooth.
• Applies moment-based analysis to derive exact conservation laws for the first, second, and third-order moments of $ u(t,x) $.
• Identifies that the $ k $-th order moment and correlation integrals scale as $ (1-t)^k $, revealing a universal scaling structure.
• Proposes a nonlocal evolution equation for $ u(t,x) $ involving the Hilbert transform, supported by verification on two closed-form solutions.
• Analyzes the implications of these conservation laws for identities involving the Hilbert transform in the context of root distributions.

Experimental results

Research questions

• RQ1 How does the root density $ u(t,x) $ of the $ (t \cdot n) $-th derivative of a high-degree polynomial evolve as $ n \to \infty $?
• RQ2 What conserved quantities govern the time evolution of $ u(t,x) $, and how do they scale with $ t $?
• RQ3 Can the evolution of $ u(t,x) $ be described by a nonlocal PDE involving the Hilbert transform, and is this consistent with known solutions?
• RQ4 What identities for the Hilbert transform emerge from the derived conservation laws?
• RQ5 What are the implications of these conservation laws for the dynamics of polynomial roots under repeated differentiation?

Key findings

• The total root density integrates to $ 1 - t $, indicating a loss of $ t $-fraction of roots as $ t $ increases from 0 to 1.
• The first moment of the root density scales linearly with $ (1 - t) $, preserving the center of mass of the initial root distribution.
• The second-order correlation integral $ \int \int u(t,x) (x - y)^2 u(t,y) \, dx \, dy $ scales as $ (1 - t)^3 $, matching the initial variance scaled by $ (1 - t)^3 $.
• An infinite family of conservation laws is established, with higher-order moments and correlation integrals scaling as $ (1 - t)^k $ for order $ k $.
• The conservation laws strongly suggest that $ u(t,x) $ evolves according to a nonlocal evolution equation involving the Hilbert transform.
• The proposed evolution equation is verified for two special closed-form solutions, supporting its plausibility and revealing new identities for the Hilbert transform.