On the Density of Coprime m-tuples over Holomorphy Rings

Abstract

Let ${\mathbb{F}}_q$ be a finite field, $F/{\mathbb{F}}_q$ be a function field of genus $g$ having full constant field ${\mathbb{F}}_q$, $\mathcal{S}$ a set of places of $F$ and $H$ the holomorphy ring of $\mathcal{S}$. In this paper we compute the density of coprime $m$-tuples of elements of $H$. As a side result, we obtain that whenever the complement of $\mathcal{S}$ is finite, the computation of the density can be reduced to the computation of the $L$-polynomial of the function field. In the rational function field case, classical results for the density of coprime $m$-tuples of polynomials are obtained as corollaries.

To appear in International Journal of Number Theory

References in corpus (1)

• The probability of rectangular unimodular matrices over $\F_q[x]$

Cited by in corpus (3)

• On the least common multiple of several random integers
• A local to global principle for densities over function fields
• On the Distribution of the Greatest Common Divisor of Gaussian Integers