Boundary zeros of stable polynomials in the unit ball

Abstract

Interpolation theory in the unit ball and semi-algebraic geometry yield explicit descriptions of the boundary zeros of stable polynomials. Given a polynomial $p\in \mathbb{C}[z_1, ...,z_n]$ that is zero-free in the unit ball and vanishes on the sphere along submanifolds of dimension at most one, we describe the boundary zeros $\mathcal{Z}(p)\cap\mathbb{S}_n$ in terms of peak sets for $A^\infty(\mathbb{B}_n)$. In particular, in the setting $n=2$, we achieve a characterization by proving that every accumulation point of $\mathcal{Z}(p)\cap\mathbb{S}_2$ lies in the relative interior of an one dimensional real analytic submanifold, and that these submanifolds form a foliation of the non-isolated part of $\mathcal{Z}(p)\cap\mathbb{S}_2$. As an application of the developed theory, we obtain a characterization of cyclic polynomials without weak essential singularities in the Dirichlet-type space $\mathcal{D}_{n-1/2}(\mathbb{B}_n)$. A theory for more general geometric settings of the boundary zeros is also developed, aiming to provide a starting point for further extensions.