Signed counts of real simple rational functions

We study the problem of counting real simple rational functions $\phi$ with prescribed ramification data (i.e. a particular class of oriented real Hurwitz numbers of genus $0$). We introduce a signed count of such functions that is invariant under change of the branch locus, thus providing a lower bound for the actual count (which does depend on such change). We prove (non-)vanishing theorems for these signed counts and study their asymptotic growth when adding further simple branch points. The approach is based on the works of Itenberg and Zvonkine (arXiv:1609.05219) which treat the polynomial case.