Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion

Kitaoka, Yoshiyuki

doi:10.3390/math13213401

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Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion

by

Yoshiyuki Kitaoka

Independent Researcher, Nagoya 468-0073, Aichi, Japan

Mathematics 2025, 13(21), 3401; https://doi.org/10.3390/math13213401 (registering DOI)

Submission received: 3 September 2025 / Revised: 26 September 2025 / Accepted: 20 October 2025 / Published: 25 October 2025

(This article belongs to the Special Issue Analytic Methods in Number Theory and Allied Fields)

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Abstract

Let

f (x)

be a monic integral polynomial of degree n and p a prime number, for which

f (x)

is fully decomposable modulo p. Let

r_{1}, \dots, r_{n}

be the roots of

f (x) mod p

with

0 \leq r_{1} \leq \dots \leq r_{n} < p

. We have conjectured that the sequence of

(r_{1}, \dots, r_{n}) / p

is uniformly distributed in some sense. We provide a clear explanation of this and generalize the Weyl criterion.

Keywords: local root of a polynomial; uniform distribution

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Kitaoka, Y. Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion. Mathematics 2025, 13, 3401. https://doi.org/10.3390/math13213401

AMA Style

Kitaoka Y. Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion. Mathematics. 2025; 13(21):3401. https://doi.org/10.3390/math13213401

Chicago/Turabian Style

Kitaoka, Yoshiyuki. 2025. "Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion" Mathematics 13, no. 21: 3401. https://doi.org/10.3390/math13213401

APA Style

Kitaoka, Y. (2025). Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion. Mathematics, 13(21), 3401. https://doi.org/10.3390/math13213401

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Notes on the Distribution of Roots Modulo a Prime of a Polynomial V: Weyl’s Criterion

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