Signed counts of real simple rational functions

TL;DR

This paper introduces a signed counting method for real simple rational functions with fixed ramification data, providing invariance under branch locus changes and establishing lower bounds and asymptotic behavior.

Contribution

It presents a new signed count invariant for real simple rational functions that remains constant under branch locus variations, extending previous polynomial case results.

Findings

Signed counts are invariant under branch locus changes.

Non-vanishing and vanishing theorems for these counts.

Asymptotic growth analyzed with additional simple branch points.

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 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.