2024/05/14 by Michael J. Mossinghoff, Mossinghoff, Michael J. · 1 citation
Mathematics · #11L40 #11Y35 #30C10 #42A05 #60G50 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical functions and polynomials #Meromorphic and Entire Functions #Number Theory (math.NT) #Primary: 11R06 #Probability (math.PR) #Secondary: 11C08
paper · pdf · doi:10.48550/arxiv.2405.08281
openalex publication_date 2024/05/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03
Mahler's problem asks for the largest possible value of the Mahler measure, normalized by the L2 norm, of a polynomial with ±1 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 Lq 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 L2 norm. Further, we show that these asymptotic values are not disturbed by a small change to make polynomials having entirely ±1 coefficients, which establishes the result on Mahler's problem. We also estimate that the limiting value of the normalized L1 norm of these polynomials exceeds 0.977, in connection with a question of Newman.