Critical Points of Polynomials With Roots on the Unit Circle

Abstract

Abstract Let ${\mathcal {L}}(u)$ be a polynomial with all its roots on the unit circle. We establish an “angle duality” theorem relating the roots and critical points of ${\mathcal {L}}(u)$. As an application, we prove that if there is a large gap between roots of ${\mathcal {L}}(u)$, then most critical points of ${\mathcal {L}}(u)$ lie close to the unit circle. We also use our result to strengthen Sendov’s well-known conjecture for these special polynomials. Finally, we prove a sharp estimate for the location of the critical point near a small gap between roots of ${\mathcal {L}}(u)$.