# What are conditions on real coefficients for zeros of a polynomial to be on the unit circle?

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.

• The first Google result for "polynomial with all roots on the unit circle" is mathpages.com/home/kmath294.htm . Please look harder for answers next time. – Qiaochu Yuan Oct 27 '09 at 16:24
• I saw that link before posting here. It has interesting information, but so far as I can tell, does not answer the question. I suspect the answer to the question is: no simple conditions on the coefficients exist. I would be happy to learn otherwise. – rita the dog Oct 27 '09 at 17:04