  We give a description of the intersection of the zero set with the unit
sphere of a zero-free polynomial in the unit ball of $\mathbb{C}^n$. This
description leads to the formulation of a conjecture regarding the
characterization of polynomials that are cyclic in Dirichlet-type spaces in the
unit ball of $\mathbb{C}^n$. Furthermore, we answer partially ascertaining
whether an arbitrary polynomial is not cyclic.
