Analízis Tanszék

Előadó: Erdélyi Tamás (Texas A&M)

Időpont és hely: 2015. 03. 18., 16 óra, H306

Littlewood polynomials are polynomials with each of their coefficients in {−1, 1}. In this talk we focus on the Mahler measure of Littlewood polynomials. A recent result establishes the expected value of the Mahler measure of Littlewood polynomials of degree n. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. The Rudin-Shapiro polynomials appear in Harold Shapiro’s 1951 thesis at MIT and are sometimes called just Shapiro polynomials. They also arise independently in a paper by Golay in 1951. They are remarkably simple to construct and are a rich source of counterexamples to possible conjectures. Despite the simplicity of their definition not much is known about the Rudin-Shapiro polynomials. It is shown in this talk that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. This settles a longstanding conjecture of a number of experts. Some consequences of this result may also be mentioned. Another example of Littlewood polynomials are the Fekete polynomials whose coefficients are Legendre symbols. Analogous results on the Mahler measure of the Fekete polynomials are also discussed.