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

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.

Revised version