Statistical distribution of roots of a polynomial modulo primes

TL;DR

This paper investigates the distribution patterns of roots of irreducible polynomials modulo primes, proposing conjectures about the behavior of sums of roots scaled by the prime, as the prime tends to infinity.

Contribution

It introduces new conjectures on the distribution of roots of polynomials modulo primes, expanding understanding of their asymptotic behavior.

Findings

Proposes conjectures on the distribution of sums of roots modulo primes.

Analyzes the asymptotic behavior of roots of polynomials modulo large primes.

Abstract

Let $f(x)=x^n+a_{n-1}x^{n-1}+\dots+a_0$ be an irreducible polynomial with integer coefficients. For a prime $p$ for which $f(x)$ is fully splitting modulo $ p$, we consider $n$ roots $r_i$ of $f(x)\equiv 0\bmod p$ with $0 \le r_1\le\dots\le r_n<p$ and propose several conjectures on the distribution of an integer $\lceil \sum_{i\in S} r_i/p\rceil$ for a subset $S$ of $\{1,\dots,n\}$ when $p\to\infty$.