Mahler's problem and Turyn polynomials

TL;DR

This paper advances Mahler's problem by analyzing Turyn polynomials, establishing a new record Mahler measure exceeding 0.95, and connecting asymptotic behaviors with random processes and coefficient modifications.

Contribution

It introduces a detailed analysis of Turyn polynomials' Mahler measures, extending random process models, and demonstrates the robustness of these measures under coefficient adjustments.

Findings

Turyn polynomials can exceed 0.95 Mahler measure of their L2 norm.

Asymptotic normalized Mahler measure and Lq norms are explicitly determined.

The results are stable under small modifications to polynomial coefficients.

Abstract

Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the $L_2$ norm, of a polynomial with $\pm1$ coefficients and large degree. We establish a new record value in this problem exceeding $0.95$ by analyzing certain Turyn polynomials, which are defined by cyclically shifting the coefficients of a Fekete polynomial by a prescribed amount. It was recently established that the distribution of values over the unit circle of Fekete polynomials of large degree is effectively modeled by a particular random point process. We extend this analysis to the Turyn polynomials, and determine expressions for the asymptotic normalized Mahler measure of these polynomials, as well as for their normalized $L_q$ norms. We also describe a number of calculations on the corresponding random processes, which indicate that the Turyn polynomials where the shift is approximately $1/4$ of the length have Mahler measure exceeding $95\%$ of their $L_2$ norm. Further, we show that these asymptotic values are not disturbed by a small change to make polynomials having entirely $\pm1$ coefficients, which establishes the result on Mahler's problem. We also estimate that the limiting value of the normalized $L_1$ norm of these polynomials exceeds $0.977$, in connection with a question of Newman.