<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>f</italic> (<italic>x</italic>) bea monicpolynomialwith integer coefficients and integers <italic>r</italic>1,..., <italic>r</italic><italic>n</italic> with 0 ≤ <italic>r</italic>1 ≤··· ≤ <italic>r</italic><italic>n &lt;p</italic> the <italic>n</italic> roots of <italic>f</italic> (<italic>x</italic>) ≡ 0mod <italic>p</italic> for a prime <italic>p</italic>. We proposed conjectures on the distribution of the point (<italic>r</italic>1<italic>/p,...,r</italic><italic>n</italic><italic>/p</italic>) in the previous papers. One aim of this paper is to revise them for a reducible polynomial <italic>f</italic> (<italic>x</italic>), and the other is to show that they imply the one-dimensional equidistribution of <italic>r</italic>1<italic>/p,...,r</italic><italic>n</italic><italic>/p</italic> for an irreducible polynomial <italic>f</italic> (<italic>x</italic>) by a geometric way.</abstract>

Let f (x) bea monicpolynomialwith integer coefficients and integers r1,..., rn with 0 ≤ r1 ≤··· ≤ rn <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r1/p,...,rn/p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r1/p,...,rn/p for an irreducible polynomial f (x) by a geometric way.

Notes on the Distribution of Roots Modulo a Prime of a Polynomial III

Uniform distribution theory

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{"article-title":"Notes on the Distribution of Roots Modulo a Prime of a Polynomial III"}

Let f (x) bea monicpolynomialwith integer coefficients and integers r1,..., rn with 0 ≤ r1 ≤··· ≤ rn <p the n roots of f (x) ≡ 0mod p for a prime p. We...

Notes on the Distribution of Roots Modulo a Prime of a Polynomial III

Publicado en línea: 24 jul 2020

Páginas: 93 - 104

Recibido: 14 feb 2020

Aceptado: 21 abr 2020

DOI: https://doi.org/10.2478/udt-2020-0005

Palabras clave
Equidistribution, polynomial, roots modulo a prime

© 2020 Yoshiyuki Kitaoka, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Notes on the Distribution of Roots Modulo a Prime of a Polynomial III

Publicado en línea: 24 jul 2020

Páginas: 93 - 104

Recibido: 14 feb 2020

Aceptado: 21 abr 2020

DOI: https://doi.org/10.2478/udt-2020-0005

Palabras claveEquidistribution, polynomial, roots modulo a prime

© 2020 Yoshiyuki Kitaoka, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Palabras clave
Equidistribution, polynomial, roots modulo a prime