Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
نویسنده
چکیده
Suppose that 2d − 2 tangent lines to the rational normal curve z 7→ (1 : z : . . . : z) 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 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.
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تاریخ انتشار 1999