An update on Fekete polynomials

Extremal properties of Littlewood polynomials (with coefficients $\pm 1$) have been extensively studied throughout the past century. Among special classes of Littlewood polynomials, particular attention has been given to so-called "Fekete polynomials" (with coefficients being Legendre symbols). Since their discovery by Dirichlet in the nineteenth century, Fekete polynomials and their extremal properties have attracted considerable attention, particularly due to their intimate connection with the putative Siegel zero and the small class number problem. I will discuss a general approach to understand the behavior of such polynomials and present various applications. This is based on a joint work with Y. Lamzouri and M. Munsch.