On sets of rational functions which locally represent all of Q

We investigate finite sets of rational functions {f1,f2,…,fr} defined over some number field K satisfying that any t0∈K is a Kp-value of one of the functions fi 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.