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Rational Functions with Real Critical Points and the B. and M. Shapiro Conjecture in real Enurative Geometry.

Le : 01/12/2010 10h30
Par : Xiao Guang WANG (LAREMA)
Lieu :
Lien web :
Résumé : This is a report on A. Ermenko and A. Garbrielov's joint work. Suppose that 2d - 2 tangent lines to the rational normal curve z (1 : z ...: z^d) 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.