Real polynomial Hurwitz numbers

TL;DR

This paper connects counts of normalized real polynomials to counts of genus zero real ramified coverings of the Riemann sphere, providing a new perspective on real Hurwitz numbers.

Contribution

It establishes a correspondence between signed counts of normalized real polynomials and genus zero real ramified coverings with specific ramification profiles.

Findings

Counts of normalized real polynomials match counts of certain real ramified coverings.

Provides a new interpretation of real Hurwitz numbers in terms of polynomial counts.

Extends previous work by Itenberg and Zvonkine on signed polynomial counts.

Abstract

We show that the signed counts of normalized real polynomials, as defined by Itenberg and Zvonkine, provide the signed counts of genus zero real ramified coverings of the Riemann sphere with a point of total ramification and several other branch points with arbitrary ramification profiles.