Real Algebraic Curves and Real Algebraic Functions

Paola Frediani

Abstract: In this paper we consider real generic holomorphic functions $f:{\cal C}\rightarrow{\bf P}^1({\bf C})$, where ${\cal C}$ is a compact connected Riemann surface of genus $g$. $f$ is said to be generic if all the critical values have multiplicity one and it is real if and only if there exists an antiholomorphic involution $\sigma$ acting on ${\cal C}$ such that for all $z$ in ${\cal C}$, $f\circ\sigma(z)=\overline{f(z)}$. It is possible to give a combinatoric description of the monodromy of the unramified covering obtained by restricting $f$ to ${\cal C}-f^{-1}(B)$, where $B$ is the set of critical values of $f$. In this paper we want to describe the topological type of the antiholomorphic involution $\sigma$ of the Riemann surface ${\cal C}$ that gives the real structure, once we know the monodromy graph of $f$. More precisely, we give a lower bound on the number of connected components of the fixed point locus of $\sigma$ in terms of the monodromy graph, in the case in which $f$ has all real critical values. Moreover, we are able to determine the exact number of the fixed components of $\sigma$ in terms of the monodromy graph, when the monodromy graph satisfies some suitable properties.

Keywords: real algebraic curves; real algebraic functions; monodromy graphs; Reidemeister--Schreier method.

Classification (MSC2000): 05C30, 57M15, 14H30.

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