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

This paper investigates the distribution of points with $p$ -adic valuation 1 for a polynomial in two variables within a random $p \times p$ square, establishing that the count follows a Poisson distribution as $p \to \infty$ under a conjecture related to the uniform distribution of a vector-valued sequence.

Original authors: Krishnan Rajkumar, Shubham

Published 2026-02-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Krishnan Rajkumar, Shubham

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about numbers. Specifically, you are looking at a giant grid of numbers (like a spreadsheet) and asking a very specific question: "How often does a certain mathematical formula produce a result that is 'deeply divisible' by a prime number?"

This paper, written by Krishnan Rajkumar and Shubham, is about finding a hidden pattern in that chaos. Here is the story of their discovery, explained without the heavy math jargon.

The Setting: The Prime Number Grid

Imagine you have a giant checkerboard. Instead of black and white squares, every square holds a number generated by a formula (like $x^3 + y^2 + 1$ ).

Now, pick a specific "prime" number, let's call it $p$ (like 7, 17, or 1013). In the world of math, we can ask: "How many times does $p$ divide the number in this square?"

If the number is 14 and $p=7$ , it divides once.
If the number is 49 and $p=7$ , it divides twice.
If the number is 343 and $p=7$ , it divides three times.

The authors are interested in squares where the number is divisible by $p$ more than once. They call this a "valuation greater than 1."

The Mystery: Is it Random or Patterned?

The authors noticed something strange.

The "Zero" Pattern: If you look for squares where the number is divisible by $p$ at least once, the pattern is very orderly. It repeats every $p$ steps, like a wallpaper design.
The "Deep" Pattern: But when you look for squares where the number is divisible by $p$ twice or more, the pattern breaks. It looks messy. It looks almost random.

They asked: "If I pick a random $p \times p$ square from this giant grid, how many 'deeply divisible' points will I find inside it?"

The Analogy: The Raindrop Game

Imagine it's raining on a giant field (the grid).

Light Rain (Divisible by $p$ ): The raindrops fall in a perfect, predictable grid pattern. You can predict exactly where they will land.
Heavy Rain (Divisible by $p^2$ ): The heavy drops only fall where the light drops already landed, but they are much rarer.

The authors wanted to know: If you take a random bucket (a $p \times p$ square) and put it on the field to catch the heavy rain, how many drops will you catch?

Sometimes you catch 0.
Sometimes you catch 1.
Sometimes you catch 2.
Rarely, you catch 3 or more.

The Big Discovery: The "Poisson" Surprise

The authors found that as the prime number $p$ gets bigger and bigger (making the grid larger and the "rain" sparser), the number of heavy drops you catch in your bucket follows a very famous statistical rule called the Poisson Distribution.

What does that mean in plain English?
It means the "messy" randomness isn't actually chaotic. It's a specific kind of randomness.

If you run this experiment a million times with a huge prime number, you will find that 1 heavy drop is the most common result.
Finding 0 drops is also very common.
Finding 2 drops is less common.
Finding 3 or more is very rare.

It's like flipping a coin a million times. You know the odds. The authors proved that this specific mathematical "rain" behaves exactly like a coin flip where the odds are perfectly balanced to create this specific bell-curve shape (the Poisson curve).

The Secret Ingredient: The "Uniformity" Guess

How did they prove this? They had to make a big guess (a Conjecture).

Imagine the "heavy rain" points are like people at a massive party.

Old thinking: Maybe these people are standing in clumps or lines.
The Authors' Guess: No, as the party gets huge, these people spread out so perfectly evenly that they look like they were placed by a random number generator. They are "uniformly distributed."

They proved that IF these points spread out evenly enough (which their computer simulations suggest they do), THEN the number of points in any random square must follow the Poisson distribution.

Why Should You Care?

This might sound like abstract math, but it's actually about understanding the hidden order in chaos.

Cryptography: Prime numbers are the backbone of internet security. Understanding how they behave in complex formulas helps us build better locks (and pick better keys).
Predictability: It shows that even when things look messy and random, there is often a deep, elegant statistical law governing them.

The Bottom Line

The authors took a complex mathematical problem about how numbers divide into prime factors, realized it looked like a random scattering of dots, and proved that this "randomness" actually follows a precise, predictable law (the Poisson distribution), provided the dots spread out evenly enough.

They didn't just count the dots; they found the rhythm in the noise.

1. Problem Statement

The paper investigates the statistical distribution of points within a random $p \times p$ square in the integer lattice $\mathbb{Z}^2$ where the $p$ -adic valuation of a given bivariate polynomial $f(x, y) \in \mathbb{Z}[x, y]$ is strictly greater than 1.

Context: While the distribution of points where $\nu_p(f(n)) > 0$ (i.e., $f(n) \equiv 0 \pmod p$ ) is well-understood via the Hasse-Weil bounds and periodicity, the behavior of points where $\nu_p(f(n)) > 1$ is more complex.
Specific Question: For a fixed polynomial $f(x, y)$ and a large prime $p$ , if one selects a random $p \times p$ square (defined by a random shift $(kp, lp)$), what is the probability distribution of the number of points $(x, y)$ in that square such that $\nu_p(f(x, y)) > 1$ ?
Hypothesis: The authors conjecture that as $p \to \infty$ , this count follows a Poisson distribution with parameter $\lambda = 1$ .

2. Methodology

The authors employ a combination of algebraic number theory, combinatorics, and probabilistic methods.

A. Indicator Random Variables

The core of the methodology involves defining indicator random variables $Y_{x,y}$ for each point $(x, y)$ on the curve $C_f(\mathbb{F}_p) = \{(x, y) \in \mathbb{F}_p^2 : f(x, y) \equiv 0 \pmod p\}$ .

$Y_{x,y}(k, l) = 1$ if $\nu_p(f(x + kp, y + lp)) > 1$ , and $0$ otherwise.
The total count in a random square is $X = \sum_{(x,y) \in C_f(\mathbb{F}_p)} Y_{x,y}$ .
The authors analyze the moments of $X$ by studying the joint distributions of these indicator variables.

B. Algebraic Analysis via Taylor Expansion

To determine when $\nu_p(f(x, y)) > 1$ , the authors utilize Taylor series expansions modulo $p^2$ .

For a point $(r, s)$ where $f(r, s) \equiv 0 \pmod p$ , the condition $\nu_p(f(r+kp, s+lp)) > 1$ reduces to a linear system involving the partial derivatives $f_x$ and $f_y$ .
They define a matrix $B_k(P)$ for a $k$ -tuple of points $P$ on the curve, containing the partial derivatives and the normalized value $f/p$ .
The rank and linear independence of the columns of this matrix determine the correlation between the indicator variables.

C. The Key Conjecture (Conjecture 1)

The proof of the main theorem relies on a specific conjecture regarding the sparsity of certain point configurations:

Conjecture 1: Let $m_k(f, p)$ be the number of element-wise distinct $k$ -tuples of points on $C_f(\mathbb{F}_p)$ such that the associated matrix $B_k(P)$ has rank 2 and its first two columns are linearly independent. The conjecture states:
$\lim_{p \to \infty} \frac{m_k(f, p)}{p^2} = 1$
This implies that "generic" $k$ -tuples of points on the curve behave in a way that makes the corresponding linear systems solvable in a specific, sparse manner, effectively decoupling the dependencies between the indicator variables in the limit.

D. Uniform Distribution Connection

The authors relate Conjecture 1 to the uniform distribution of a vector-valued sequence $F_p$ in $\mathbb{F}_p^3$ , where $F_p = \{(f_x, f_y, f/p) \mid (x, y) \in C_f(\mathbb{F}_p)\}$ . They propose that the discrepancy of this sequence tends to zero, which would imply Conjecture 1.

3. Key Contributions

Formulation of the Poisson Limit: The paper establishes that the number of points with valuation $>1$ in a random $p \times p$ square converges to a Poisson distribution with $\lambda=1$ , provided Conjecture 1 holds.
Generalization of Valuation Trees: The work extends previous results on $p$ -adic valuations (often studied for single-variable polynomials or specific cases like $n^2-a$ ) to general bivariate polynomials using a "valuation tree" framework generalized to two dimensions.
Algebraic-Probabilistic Bridge: The authors successfully bridge the gap between the algebraic geometry of the curve $C_f(\mathbb{F}_p)$ (via the rank of the matrix $B_k$ ) and the probabilistic independence of the valuation events.
Moment Method Proof: They prove the convergence in distribution by showing that the $k$ -th moments of the random variable $X$ converge to the $k$ -th moments of the Poisson(1) distribution. This relies on the Stirling numbers of the second kind and the asymptotic behavior of the sums of joint expectations.

4. Main Results

Theorem 1: Let $f(x, y) \in \mathbb{Z}[x, y]$ be a polynomial such that the system $f \equiv 0, f_x \equiv 0, f_y \equiv 0 \pmod p$ has no solutions for sufficiently large $p$ . If Conjecture 1 holds, then the random variable $X$ (count of points with $\nu_p > 1$ in a random $p \times p$ square) converges in distribution to a Poisson random variable with parameter $\lambda = 1$ as $p \to \infty$ .
Corollary 4: Assuming Conjecture 1, the limit of the sum of joint expectations of $k$ distinct indicator variables is 1, which is the necessary condition for the moments to match the Poisson distribution.
Numerical Evidence: The paper provides numerical data (Table 1 and Figure 3) supporting Conjecture 1 and Conjecture 2 (uniform distribution). The data shows that the ratio $m_k(f, p)/p^2$ approaches 1 and the discrepancy measures decay as $O(p^{-1/2})$ , consistent with theoretical bounds.

5. Significance

Theoretical Insight: This work deepens the understanding of the local distribution of integer points on algebraic curves with respect to $p$ -adic valuations. It suggests that despite the algebraic constraints, the "higher valuation" points behave statistically like a random process (Poisson) in the limit.
Methodological Advancement: The introduction of the matrix $B_k(P)$ and the analysis of its rank provides a new tool for studying the correlation of $p$ -adic valuations in multivariate settings.
Future Directions: The paper highlights that the result depends on a conjecture regarding uniform distribution. Proving Conjecture 1 (or the related uniform distribution conjectures) would solidify the result. The authors also note that extending this to polynomials in more than two variables is a significant open problem for future research.

In summary, the paper presents a compelling argument that the "rare" events of high $p$ -adic valuation for bivariate polynomials are distributed according to a Poisson law, linking deep algebraic properties of curves over finite fields to probabilistic limit theorems.