Question

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.

Answer 1

Edited answer (see below for my original, less useful reply): Since your polynomial p(z) of degree 2d is palindromic, rewrite it as z^d q(z+1/z) for some polynomial q(x) of degree d. Then p(z) has 2d roots on the circle if and only if q(x) has d real roots in the interval [-2,2]. (Equivalently, q(x-2) should have d nonnegative real roots, while q(x+2) should have no positive roots.) Now you can try to extract information using e.g. Sturm sequences to try to count real roots in that interval.

I had previously posted: Since the real line parameterizes the circle (minus a point), you can transform your problem into counting the real zeros of an associated real polynomial. (See p. 182 of Rodriguez Villegas's book "Experimental Number Theory" for details; this is viewable in Google Books.)

There certainly have to exist necessary & sufficient criteria for all the zeros of a real polynomial to be real, involving rather complicated inequalities in the coefficients of the polynomial, generalizing the condition b^2-4ac >= 0 for quadratics; or for a specific polynomial you can try a real-root-counting algorithm (see Sturm's theorem on Wikipedia).

Answer 2

See paper Palindrome-Polynomials with Roots on the Unit Circle by John Konvalina and Valentin Matache