On Average Modulus of Random Polynomials Over a Unit Circle and Disc

TL;DR

This paper investigates the average and maximum modulus of random polynomials with Gaussian and uniform coefficients on the unit circle and disc, providing new probabilistic bounds and insights.

Contribution

It introduces novel bounds for the maximum modulus of random polynomials and applies probabilistic methods to analyze their norms, advancing understanding in this area.

Findings

Derived bounds for maximum modulus using Markov inequality

Analyzed likelihood of exceeding specific modulus thresholds

Proposed new directions for studying norms of random polynomials

Abstract

This article presents some interesting and novel results concerning the average modulus of random polynomials on the unit circle and the unit disc, with coefficients distributed as standard normal variates. The paper also introduces new results concerning the bounds of the maximum modulus of random polynomials with coefficients distributed as independently as Gaussian and uniform variates, utilizing probability principles to derive findings about the likelihood of the maximum modulus exceeding a specific threshold, using Markov inequality as the primary probabilistic tool. These findings and the approach can potentially initiate the study of a rich class of problems concerning the norms of random polynomials.