On the Density of Coprime m-tuples over Holomorphy Rings

TL;DR

This paper calculates the density of coprime m-tuples over holomorphy rings in function fields, linking the problem to the L-polynomial when the set of places is finite, and recovers classical polynomial results.

Contribution

It provides a general formula for the density of coprime m-tuples over holomorphy rings in function fields, extending classical polynomial results.

Findings

Density expressed via L-polynomial for finite complements of S

Recovers classical polynomial density results as special cases

Connects density computation to algebraic properties of the function field

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.