Translator Disclaimer
VOL. 49 | 2007 The probability of two $\mathbb{F}_q$-polynomials to be coprime
Hiroshi Sugita, Satoshi Takanobu

Editor(s) Shigeki Akiyama, Kohji Matsumoto, Leo Murata, Hiroshi Sugita

Abstract

By means of the adelic compactification $\widehat{R}$ of the polynomial ring $R := \mathbb{F}_q [x]$, $q$ being a prime, we give a probabilistic proof to a density theorem: $$ \frac{\# \{(m, n) \in \{0, 1, \dots, N-1\}^2\ ;\ \varphi_m \text{ and }\varphi_n \text{ are coprime}\}}{N^2} \to \frac{q-1}{q}, $$ as $N \to \infty$, for a suitable enumeration $\{\varphi_n\}_{n=0}^{\infty}$ of $R$. Then establishing a maximal ergodic inequality for the family of shifts $\{\widehat{R} \ni f \mapsto f + \varphi_n \in \widehat{R}\}_{n=0}^{\infty}$, we prove a strong law of large numbers as an extension of the density theorem.

Information

Published: 1 January 2007
First available in Project Euclid: 27 January 2019

zbMATH: 1154.60006
MathSciNet: MR2405615

Digital Object Identifier: 10.2969/aspm/04910455

Subjects:
Primary: 60B10
Secondary: 60B15, 60F15

Rights: Copyright © 2007 Mathematical Society of Japan

PROCEEDINGS ARTICLE
24 PAGES


SHARE
Back to Top