Interpolation on real Riemann surfaces

2010-08-01T09:14:14Z

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?

Answer by Dmitri for Interpolation on real Riemann surfaces

2010-08-01T21:18:53Z

Let me give a version of the question in the comment:

Let $X$ be a curve of genus $g$ with a real separting involution, and conisder the map $Sym^n(X_+)\to Jac^n(X)$.

For wich $n$ is this map surjective? Or, in other words, what is the minimal number of poles of a meromorphic function with poles in $X_+$ that garanties that zeros can happen at any collection of points?

This sounlds like a very nice question. In the case $g=1$ you can always take $n=2$. Also for any $g$ you sould take $n>g$ because $Sym^g(X)$ maps to $Jac^g(X)$ with degree $1$.

Added. The notation $Sym^n(X)$ means the symmetric power of $X$. Let me explain also why what is above is a reformulation of the original question. Indeed, a divisor $\sum_i x_i-\sum_i y_i$ on $X$ is a divisor of a meromorfic function iff it represent zero in $Jac^0(X)$. So if we want to chose arbitraly zeros $x_i$ of a meromorphic function $f$ keeping the poles $y_i$ in $X_+$ it is enouth to know that $\sum_i y_i$ can take any value in $Jac^n(X)$ (to cancel the point $\sum_i x_i$). This is eactly the condition that $Sym^n(X_+)\to Jac^n(X)$ is surjective.

Interpolation on real Riemann surfaces - MathOverflow

Interpolation on real Riemann surfaces

Answer by Dmitri for Interpolation on real Riemann surfaces