The Density of Tuples Restricted by Relatively r-Prime Conditions

TL;DR

This paper introduces a generalized concept of j-wise relative r-primality in Dedekind domains using hypergraph structures, extending natural density results to algebraic integer rings and polynomial rings over finite fields.

Contribution

It defines j-wise relative r-primality with hypergraph constraints, broadening the scope of density results beyond traditional prime conditions.

Findings

Generalized density formulas for Dedekind domains.

Extended results to rings of algebraic integers.

Applied to polynomial rings over finite fields.

Abstract

In order to consider j-wise relative r-primality conditions that do not necessarily require all j-tuples of elements in a Dedekind domain to be relatively r-prime, we define the notion of j-wise relative r-primality with respect to a fixed j-uniform hypergraph H. This allows us to provide further generalisations to several results on natural densities not only for a ring of algebraic integers, but also for the ring F_q[x].