I present a recent work with B\´en\´eteau, Khavinson, Liaw and Simanek where we study the structure of the zeros of polynomials appearing in the study of cyclicity in Hilbert spaces of analytic functions. We find the minimum possible modulus of occurring zeros via a nonlinear extremal problem associated with norms of Jacobi matrices. We examine global properties of these zeros and prove Jentzsch-type theorems describing where they accumulate.