## Advanced Studies in Pure Mathematics

### The probability of two $\mathbb{F}_q$-polynomials to be coprime

#### 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.

#### Article information

Dates
Revised: 20 March 2006
First available in Project Euclid: 27 January 2019

Sugita, Hiroshi; Takanobu, Satoshi. The probability of two $\mathbb{F}_q$-polynomials to be coprime. Probability and Number Theory — Kanazawa 2005, 455--478, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04910455. https://projecteuclid.org/euclid.aspm/1548550910