Boundary zeros of stable polynomials in the unit ball

Abstract

We study polynomials in several complex variables that are stable (i.e. they have no zeros in the unit ball $\mathbb{B}_n$), but vanish on the unit sphere along submanifolds of dimension at most one. We characterize the zeros of such polynomials on the unit sphere in terms of peak sets for the space $A^\infty(\mathbb{B}_n).$ Furthermore, we explicitly construct a polynomial in $\mathbb{C}^3$ that provides a negative answer to the question of whether the equivalent conditions of this characterization universally hold for all stable polynomials. As an application of the developed theory, we obtain a characterization of a certain class of cyclic polynomials in the Dirichlet-type space $D_{n-\frac{1}{2}}(\mathbb{B}_n).$ Next, we examine polynomials whose local zero set, in a neighborhood of points on the unit sphere, coincides with the graph of a holomorphic function. In this particular case, we achieve a characterization of the set of zeros lying on the unit sphere whenever the zero set has local maximum dimension $n-1$. Lastly, we discuss potential generalizations and open problems.