Hurwitz numbers for real polynomials

TL;DR

This paper introduces real analogs of polynomial Hurwitz numbers, establishing their invariance under branch point order, analyzing generating series, and deriving asymptotic behavior as polynomial degree increases.

Contribution

It defines and computes real polynomial Hurwitz numbers with fixed ramification, showing their invariance and analyzing their generating series and asymptotics.

Findings

Number of real polynomials is independent of branch point order when counted with sign.

Derived conditions for the vanishing and nonvanishing of generating series.

Established a logarithmic asymptotic for the invariants as degree grows.

Abstract

We consider the problem of defining and computing real analogs of polynomial Hurwitz numbers, in other words, the problem of counting properly normalized real polynomials with fixed ramification profiles over real branch points. We show that, provided the polynomials are counted with an appropriate sign, their number does not depend on the order of the branch points on the real line. We study generating series for the invariants thus obtained, determine necessary and sufficient conditions for the vanishing and nonvanishing of these generating series, and obtain a logarithmic asymptotic for the invariants as the degree of the polynomials tends to infinity.