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

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