On sets of rational functions which locally represent all of $\mathbb{Q}$

TL;DR

This paper studies finite sets of rational functions over number fields that collectively take on every value in the field at almost all primes, providing necessary conditions and examples related to intersective polynomials and arithmetically exceptional functions.

Contribution

It establishes necessary conditions on the shape of such rational functions and connects the problem to intersective polynomials and arithmetically exceptional functions.

Findings

Derived strong necessary conditions for the functions' shapes.

Provided concrete examples illustrating near-sufficiency of these conditions.

Linked the problem to intersective polynomials and arithmetically exceptional functions.

Abstract

We investigate finite sets of rational functions $\{ f_{1},f_{2}, \dots, f_{r} \}$ defined over some number field $K$ satisfying that any $t_{0} \in K$ is a $K_{p}$-value of one of the functions $f_{i}$ for almost all primes $p$ of $K$. We give strong necessary conditions on the shape of functions appearing in a minimal set with this property, as well as numerous concrete examples showing that these necessary conditions are in a way also close to sufficient. We connect the problem to well-studied concepts such as intersective polynomials and arithmetically exceptional functions.