We use methods from potential theory and harmonic analysis to show non-cyclicity of polynomials on a polydisc whose zero set meets the distinguished boundary along a hypersurface. We also generalize methods used for proving cyclicity for polynomials in two variables with small zero sets to arbitrary dimension. In doing so, we show that in higher dimension, the cyclicity properties of a function do not only depend on the codimension, but also on the orientation of the zero set. Furthermore, we illustrate our results by studying a special class of polynomials. Finally, we use methods from potential theory to prove that our estimates for non-cyclicity are in fact sharp.