Rodrigues' descendants of a polynomial and Boutroux curves

TL;DR

This paper analyzes the asymptotic root distribution of derivatives of polynomial powers, revealing connections to Boutroux curves and providing explicit harmonic functions and differential equations.

Contribution

It introduces a novel asymptotic description of polynomial derivatives using harmonic functions linked to Boutroux curves, expanding understanding of their geometric and analytic properties.

Findings

Asymptotic root distribution described by a harmonic function

Identification of a Boutroux curve associated with the polynomial sequence

Derivation of a differential equation satisfied by the polynomial derivatives

Abstract

Motivated by the classical Rodrigues' formula, we study the root asymptotic of the polynomial sequence $$R_{[\alpha n],n,P}(z)=\frac{d^{[\alpha n]}P^n(z)}{dz^{[\alpha n]}}, n= 0,1,\dots$$ where ${P(z)}$ is a fixed univariate polynomial, $\alpha $ is a fixed positive number smaller than deg $P$, and $[\alpha n]$ stands for the integer part of $\alpha n$. Our description of this asymptotic is expressed in terms of an explicit harmonic function uniquely determined by the plane rational curve emerging from the application of the saddle point method to the integral representation of the latter polynomials using Cauchy's formula for higher derivatives. As a consequence of our method, we conclude that this curve is birationally equivalent to the zero locus of the bivariate algebraic equation satisfied by the Cauchy transform of the asymptotic root-counting measure for the latter polynomial sequence. We show that this harmonic function is also associated with an abelian differential having only purely imaginary periods and the latter plane curve belongs to the class of Boutroux curves initially introduced by Bertola. As an additional relevant piece of information, we derive a linear ordinary differential equation satisfied by $\{R_{[\alpha n],n,P}(z)\}$ as well as higher derivatives of powers of more general functions.