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arXiv · 2411.05366

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

Krishnan Rajkumar·Shubham
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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.

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Krishnan Rajkumar, Shubham. 2024-11-08. On the Distribution of Points of Valuation 1 for a Polynomial in Two Variables. https://arxiv.org/abs/2411.05366

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