RATIONAL FUNCTIONS WITH REAL CRITICAL POINTS AND

THE B. AND M. SHAPIRO CONJECTURE IN REAL ENUMERATIVE GEOMETRY

ALEX EREMENKO ANDREI GABRIELOV

Abstract. Suppose that 2d−2 tangent lines to the rational normal curve of degree d, z ↦ (1 : z : …: 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.