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Background: Generalizing the notion of upper half plane to compact Riemann surfaces:

Suppose $p(x,y) \in \mathbb{R}[x,y]$ is a polynomial in 2 variables with real coefficients, defining a smooth complex plane algebraic curve $C_0 = {(x,y) \in \mathbb{C}^2:p(x,y)=0}$. Let $C$ be the projective closure of $C_0$ in $P^2\mathbb{C}$, and assume that $C$ is also smooth. Since $C$ is defined over the real numbers, it comes equipped with an involution $\sigma:C\rightarrow C$, $\sigma(x,y) = (\overline{x},\overline{y})$. Denote by $X$ the compact Riemann surface associated to $C$, and let $X_\mathbb{R}$ be the set of fixed points of $\sigma$.

If the space $X \ - X_\mathbb{R}$ has exactly two connected components, then $X$ is called a real compact Riemann surface of dividing type, and the two connected components are denoted by $X_+$ and $X_{-}$ (the decision between "the positive half plane" and "the negative half plane" arbitrarily).

And finally, to the question:

I am given a real compact Riemann surface of dividing type $X$, and interested in interpolation problems of meromorphic functions with conditions such as "all the poles of $f$ lie in the upper half plane". Does anybody knows of any previous work in the area? Any known techniques to relate these topological and algebraic constructions?

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# Interpolation on real Riemann surfaces

Background: Generalizing the notion of upper half plane to compact Riemann surfaces:

Suppose $p(x,y) \in \mathbb{R}[x,y]$ is a polynomial in 2 variables with real coefficients, defining a smooth complex plane algebraic curve $C_0 = {(x,y) \in \mathbb{C}^2:p(x,y)=0}$. Let $C$ be the projective closure of $C_0$ in $P^2\mathbb{C}$, and assume that $C$ is also smooth. Since $C$ is defined over the real numbers, it comes equipped with an involution $\sigma:C\rightarrow C$, $\sigma(x,y) = (\overline{x},\overline{y})$. Denote by $X$ the compact Riemann surface associated to $C$, and let $X_\mathbb{R}$ be the set of fixed points of $\sigma$.

If the space $X \ X_\mathbb{R}$ has exactly two connected components, then $X$ is called a real compact Riemann surface of dividing type, and the two connected components are denoted by $X_+$ and $X_{-}$ (the decision between "the positive half plane" and "the negative half plane" arbitrarily).

And finally, to the question:

I am given a real compact Riemann surface of dividing type $X$, and interested in interpolation problems of meromorphic functions with conditions such as "all the poles of $f$ lie in the upper half plane". Does anybody knows of any previous work in the area? Any known techniques to relate these topological and algebraic constructions?