Infinite intersection of maximal ideals in two variable polynomial ring

Question

Consider the polynomial ring $\mathbf{C}[x,y]$ in two variables. It is a standard fact that this ring is Jacobson and hence the intersection of all its maximal ideals is zero. I am interested to know if there is a criterion in determining when an infinite intersection of distinct maximal ideals of this ring is zero. In particular, in my research I've found myself needing to determine whether the intersection of the form $$\bigcap_{n\in\mathbf{Z}_{\geq1}} \left(x-\frac{2}{p^{n-\frac{1}{2}}},y-\frac{1}{p^{2n}}\right)$$ is zero or not (here $p$ denotes an odd prime). It feels like it should but I just cant seem to be able to come up with an argument to prove or disprove it.

Answer 1

It seems that the non-zero polynomial $ 4py-x^2$ is in the intersection.