Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
Alex Eremenko and Andrei Gabrielov
Suppose that $2d-2$ tangent lines to the rational normal curve of degree $d$, $z\mapsto (1:z:\ldots:z^d)$ in $d$-dimensional projective space are given. It was known that the number of codimension $2$ subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the $d$-th Catalan number. We prove that for real tangent lines, all these codimension $2$ subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result:If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.