Fekete polynomials play an essential role in studying special values of L-functions of quadratic fields. In previous joint work with Jan Minac and Nguyen Duy Tan, we investigated the arithmetic of Fekete polynomials attached to a quadratic Dirichlet character with a prime conductor. In this talk, we will introduce the generalized Fekete polynomial associated with a general quadratic Dirichlet character. We will then determine their cyclotomic and non-cyclotomic factors. Based on extensive numerical data, we will show that the Galois group of these generalized Fekete polynomials seems to follow a rather elegant pattern. Time permitting, we will explain how to utilize this circle of ideas to construct a new class of (Ramanujan) generalized Paley graphs.

Bio: Tung T. Nguyen is a Vietnamese mathematician. He earned his doctorate from the University of Chicago under the supervision of Professor Kazuya Kato. He is currently a postdoctoral associate in Muller's lab at Western University. His primary research interests are in computational/algebraic number theory and nonlinear dynamics. He has also worked on several side projects on spectral graph theory, non-commutative ring theory, and Galois modules.