On the Distribution of Points of Valuation 1 for a Polynomial in Two Variables

We investigate the variation in the total number of points in a random $p\times p$ square in ${\mathbb{Z}}^2$ where the $p$-adic valuation of a given polynomial in two variables is precisely $1$. We establish that this quantity follows a Poisson distribution as $p\to \mathrm{\infty }$ under a certain conjecture. We also relate this conjecture to certain uniform distribution properties of a vector valued sequence.