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

Eremenko, A.; Gabrielov, A.

Ann. Math. (2) 155, No. 1, 105-129 (2002).

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)

Keywords:

number of codimension 2 subspaces; Catalan number; rational function

Citations:

Zbl 0997.14015