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.
Categories: math.NT