First steps towards total reality of meromorphic functions

Ekedahl, T.; Shapiro, B.; Shapiro, M.

First steps towards total reality of meromorphic functions

Authors

Ekedahl, T.

Shapiro, B.

Shapiro, M.

Description

It was earlier conjectured by the second and the third authors that any rational curve $g:{\mathbb C}P^1\to {\mathbb C}P^n$ such that the inverse images of all its flattening points lie on the real line ${\mathbb R}P^1\subset {\mathbb C}P^1$ is real algebraic up to a linear fractional transformation of the image ${\mathbb C}P^n$. (By a flattening point $p$ on $g$ we mean a point at which the Frenet $n$-frame $(g',g'',...,g^{(n)})$ is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees $2,3$ and several other cases.
10 pages, 1 figure