Rodrigues' descendants of a polynomial and Boutroux curves
Rikard B{\o}gvad, Christian H\"agg, Boris Shapiro

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 where is a fixed univariate polynomial, is a fixed positive number smaller than deg , and stands for the integer part of . 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…
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Taxonomy
TopicsPolynomial and algebraic computation · Nonlinear Waves and Solitons · Mathematical functions and polynomials
