## Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry.(English)Zbl 0997.14015

Summary: Suppose that $$2d-2$$ tangent lines to the rational normal curve $$z\mapsto (1:z: \cdots:z^d)$$ 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 $$d$$-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 [see the following review Zbl 0997.14015]. 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.

### MSC:

 14N15 Classical problems, Schubert calculus 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14P99 Real algebraic and real-analytic geometry 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)

Zbl 0997.14015
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