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

Krishnan Rajkumar, Shubham

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.

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

Abstract

We investigate the variation in the total number of points in a random square in where the -adic valuation of a given polynomial in two variables is precisely . We establish that this quantity follows a Poisson distribution as under a certain conjecture. We also relate this conjecture to certain uniform distribution properties of a vector valued sequence.

Paper Structure

This paper contains 5 sections, 12 theorems, 41 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Indicator random variables for valuation greater than $1$
  4. Uniform distribution results
  5. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Let $f(x,y)\in\mathbb{Z}[x,y]$ be a polynomial with no solutions to the set of equations $f(x,y)\equiv 0 \pmod p, f_x(x,y)\equiv 0 \pmod p, \text{ and } f_y(x,y)\equiv 0 \pmod p$ for all sufficiently large prime numbers $p$. Let $X$ be the random variable defined in Xdefn, and assume that Conjecture

Figures (3)

  • Figure 1: Scatter Plot of $(x, y)$ satisfying: Upper Left: $\nu_p(x^3 + y^3 + x^2 y + y + 1) = 1$, for $p = 17$; Upper Right: $\nu_p(x^3 + y^2 x + xy + x + y + 1) = 1$, for $p = 19$; Lower Left: $\nu_p(x^3 + xy + x + y + 1) = 1$, for $p = 23$; Lower Right: $\nu_p(y^2 x + xy + x + y + 1) = 1$, for $p = 29$.
  • Figure 2: Comparison of the Poisson distribution ($\lambda=1$) with the distribution of $X$ for $f(x,y)=x^3+y^2+x y+1$ and the prime numbers, Upper Left: $p = 211$; Upper Right: $p = 503$; Lower Left: $p = 1013$; Lower Right: $p = 1033$.
  • Figure 3: Left: Plot of $m_{2}(f,p)/p^2$ versus $p$; Right: Plot of $m_{3}(f,p)/p^2$ versus $p$ for the polynomial $f(x,y)=x^3+y^2+x y+1$.

Theorems & Definitions (24)

  • Conjecture 1
  • Theorem 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • Corollary 2
  • Corollary 3
  • ...and 14 more