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

2019/05/07 by Yoshiyuki Kitaoka, Kitaoka, Yoshiyuki
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1905.02364

openalex publication_date 2019/05/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∈ ℤ, f(r_i) ≡ 0 \bmod p and 0 ≤ r₁ ≤ r₂ ≤ … ≤ r_n < p. With numerical data, we propose a conjecture on the distribution of (r₁/p,…,r_n/p), which is a new kind of equi-distribution, and a conjecture of the distribution of (r₁,…,r_n) which satisfies r_i ≡ R_i \bmod L for given natural numbers L,R₁,…,R_n, which is nothing but Dirichlet's theorem on an arithmetic progression in the case n = 1.

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