On the polynomials with all their zeros on the unit circle (Q1892235)

The author shows that a polynomial has the title property if and only if it has the form \(z^ m q(z) + e^{i \theta} q^* (z)\), where \(q(z)\) is a polynomial with all zeros within the closed unit disc and \(q^* (z)\) is its reciprocal polynomial. From this he derives sufficient conditions on the coefficients of a real polynomial (loosely, symmetry and zigzag) that its zeros lie on the unit circle. Examples are given.