My complex analysis is decades in the rear view mirror. Perhaps someone here can help. I am looking for necessary and sufficient conditions on the coefficients of of a real polynomial of one complex variable (i.e. an element of ℝ[z]) so that all of its zeros will be on the unit circle. Clearly it is necessary that the polynomial is palindromic, but that is not sufficient. My question arose, after I discovered that if r,s are both real numbers then the polynomial p(z) =

has all its roots on the unit circle. Since I had not seen anything like this before, I wondered if there were perhaps just some conditions on the coefficients one could check.