Monodromies of generic real algebraic functions (Q1817554)

Consider real holomorphic functions \(f:{\mathcal C}\to\mathbb{P}^1\) (where \({\mathcal C}\) is a compact connected Riemann surface of genus \(g)\) such that \(f\) has at most one non-generic critical value. The monodromies of such functions can be described in terms of labeled graphs. The author gives necessary and sufficient conditions for a graph to be the monodromy graph of a real algebraic function \(f:C\to\mathbb{P}^1\) of a given degree having at most one non-generic critical value. The author also proves that the number of connected components of the Hurwitz space of complex algebraic functions whose critical values have distinct absolute values is equal to the number of monodromy graphs of real algebraic functions whose critical values are real and of multiplicity 1.