Colloquium

A SURVEY ON SOME RECENT PROGRESS IN THE

STUDY OF THE RUDIN-SHAPIRO POLYNOMIALS

 Professor Tamάs Erdélyi

Department of Mathematics

Texas A&M University, College Station

 Abstract: Littlewood polynomials are polynomials with each of their coefficients in {−1, 1}. 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 H. 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 examples and

counterexamples. Despite the simplicity of their definition not much is known about the Rudin-Shapiro polynomials. In this talk a sort of survey is given on some recent progress in the study of Rudin-Shapiro polynomials. One of the highlights of the talk is an asymptotic formula, conjectured by B. Saffari in 1985, for the Mahler measure of the Rudin-Shapiro polynomials on the unit circle. An essential tool in the proof of this formula is another longstanding conjecture of B. Saffari and H. Montgomery, proved recently by B. Rodgers, on the value distribution of the Rudin-Shapiro polynomials on the unit circle. Some new results on the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle are also discussed. The talk concludes with some simple-looking but open questions.