Abstract
We study the problem of counting real simple rational functions $$\varphi $$ 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 which is independent of the position of the branch points, thus providing a lower bound for the actual count (which does depend on the position). 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 Itenberg and Zvonkine (Comment Math Helv 93(2), 441–474, 2018) which treats the polynomial case.
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