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

TL;DR

This paper studies the distribution of points where a polynomial in two variables has a specific p-adic valuation, showing it follows a Poisson distribution under certain conditions as the size grows large.

Contribution

It establishes a link between the distribution of valuation points and Poisson behavior, proposing a conjecture related to uniform distribution properties.

Findings

Number of valuation points follows a Poisson distribution asymptotically.

Conjecture connects valuation distribution to uniform distribution of sequences.

Provides theoretical insight into p-adic valuation patterns in polynomial points.

Abstract

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 $p\rightarrow\infty$ under a certain conjecture. We also relate this conjecture to certain uniform distribution properties of a vector valued sequence.