2004/05/11 by A. Eremenko, A. Gabrielov · 2 citations
Mathematics · #math.AG #math.CV
published as Ann. of Math. (2) 155 (2002), no. 1, 105--129 · 25 pages, published version
arxiv created 2004/05/11 · arxiv updated 2009/12/01
Suppose that 2d-2 tangent lines to the rational normal curve z↦ (1 : z : ... : zd) in d-dimensional complex 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 dth 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.