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

Yoshiyuki Kitaoka

paper

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

arXiv:1905.02364

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 $r_{i} \in Z$ , $f (r_{i}) \equiv 0 mod p$ and $0 \leq r_{1} \leq r_{2} \leq \dots \leq r_{n} < p$ . 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 $r_{i} \equiv R_{i} mod L$ for given natural numbers $L, R_{1}, \dots, R_{n}$ , which is nothing but Dirichlet's theorem on an arithmetic progression in the case $n = 1$ .