Advanced Studies in Pure Mathematics

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

Hiroshi Sugita and Satoshi Takanobu

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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

Probability and Number Theory — Kanazawa 2005, S. Akiyama, K. Matsumoto, L. Murata and H. Sugita, eds. (Tokyo: Mathematical Society of Japan, 2007), 455-478

Received: 17 February 2006
Revised: 20 March 2006
First available in Project Euclid: 27 January 2019

Permanent link to this document euclid.aspm/1548550910

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60F15: Strong theorems


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.

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