# Conjectures on the distribution of roots modulo a prime of a polynomial

## TL;DR

This paper proposes conjectures on the distribution of polynomial roots modulo primes, suggesting a new form of equi-distribution and relating to Dirichlet's theorem for roots in arithmetic progressions.

## Contribution

It introduces new conjectures on the distribution patterns of polynomial roots modulo primes, supported by numerical data and connecting to classical number theory.

## Key findings

- Numerical data supports the conjectured distribution patterns.
- Proposes a new type of equi-distribution for roots modulo primes.
- Links root distribution to Dirichlet's theorem in special cases.

## Abstract

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With numerical data, we propose a conjecture on the distribution of $(r_1/p,\dots,r_n/p)$, which is a new kind of equi-distribution, and a conjecture of the distribution of $(r_1,\dots,r_n)$ which satisfies $r_i \equiv R_i \bmod L$ for given natural numbers $L,R_1,\dots,R_n$, which is nothing but Dirichlet's theorem on an arithmetic progression in the case $n = 1$.