The asymptotic distance between an ultraflat unimodular polynomial and its conjugate reciprocal

TL;DR

This paper investigates the asymptotic behavior of the distance between ultraflat unimodular polynomials and their conjugate reciprocals, revealing precise growth rates and integral estimates as the degree increases.

Contribution

It provides new asymptotic formulas for the distance between ultraflat polynomials and their conjugate reciprocals, including integral estimates for their derivatives, advancing understanding of their geometric properties.

Findings

Asymptotic formula for the integral of the difference between P_n and P_n^*

Asymptotic estimate for the integral of the derivatives difference

Reproves some classical results with new methods

Abstract

Let $${\mathcal K}_n := \left\{p_n: p_n(z) = \sum_{k=0}^n{a_k z^k}, \enspace a_k \in {\mathbb C}\,,\enspace |a_k| = 1 \right\}\,.$$ A sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$ is called ultraflat if $(n + 1)^{-1/2}|P_n(e^{it})|$ converge to $1$ uniformly in $t \in {\mathbb R}$. In this paper we prove that $$\frac{1}{2\pi} \int_0^{2\pi}{\left| (P_n - P_n^*)(e^{it}) \right|^q \, dt} \sim \frac{{2}^q \Gamma \left(\frac{q+1}{2} \right)}{\Gamma \left(\frac q2 + 1 \right) \sqrt{\pi}} \,\, n^{q/2}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$ and for every $q \in (0,\infty)$, where $P_n^*$ is the conjugate reciprocal polynomial associated with $P_n$, $\Gamma$ is the usual gamma function, and the $\sim$ symbol means that the ratio of the left and right hand sides converges to $1$ as $n \rightarrow \infty$. Another highlight of the paper states that $$\frac{1}{2\pi}\int_0^{2\pi}{\left| (P_n^\prime - P_n^{*\prime})(e^{it}) \right|^2 \, dt} \sim \frac{2n^3}{3}$$ for every ultraflat sequence $(P_n)$ of polynomials $P_n \in {\mathcal K}_n$. We prove a few other new results and reprove some interesting old results as well.