How to count the number of zeros that a polynomial has on the unit circle?

Abstract

The classical problem of counting the number of real zeros of a real polynomial was solved a long time ago by Sturm. The analogous problem of counting the number of zeros that a polynomial has on the unit circle is, however, still an open problem. In this paper, we show that the second problem can be reduced to the first one through the use of a suitable pair of M\"obius transformations - often called Cayley transformations - that have the property of mapping the unit circle onto the real line and vice versa. Although the method applies to arbitrary complex polynomials, we discuss in detail several classes of polynomials with symmetric zeros as, for instance, the cases of self-conjugate, self-adjoint, self-inversive, self-reciprocal and skew-reciprocal polynomials. Finally, an application of this method to Salem polynomials and to polynomials with small Mahler measure is also discussed.