Monodromies of generic real algebraic functions

In this paper we give a combinatorial description of the monodromies of real generic (non constant) holomorphic functions f:C→P1(C), where C is a compact connected Riemann surface of genus g. The monodromy of the branched covering of P1(C) given by such a function f can be described by means of a graph with labeled edges. In this paper we describe completly these graphs in the real generic case (i.e., when all the critical values of f have multiplicity one) and also in the case in which ∞ is the only non generic critical value. We are also able to compute the number of such graphs in the case in which the functions are real generic, of degree 3 and the genus varies. Finally we generalize a result obtained in the polynomial case together with F. Catanese, namely we prove that the number of connected components of the Hurwitz space of complex lemniscate generic algebraic functions (i.e., functions whose critical values have distinct absolute values) is equal to the number of monodromy graphs of real generic algebraic functions whose critical values are all real.