Non-cyclicity and polynomials in Dirichlet-type spaces of the unit ball

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.