Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
By Alexandre Eremenko, Andrei Gabrielov. 2004
142 citations recorded by the source corpus.
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 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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6fdf8f593689a042a9823729e2a5838c72cda8fd
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
- V. Tarasov
- A. Varchenko
- E. Mukhin
- Alexandre Eremenko
- Valerio Toledano Laredo
- Andrei Gabrielov
- P. Boalch
In Mathematical Physics
Citations in the corpus
- The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz (2005, methodology): “The assignment V 7→ WrV defines a finite morphism π : G(r + 1, d) → P(C(r+1)(d−r)[x]), see, for example, [S2, EG1 ].”
- Metrics of positive curvature with conic singularities on the sphere (2002, background): “On the other hand, if all singularities belong to the real line, then all solutions of this problem are symmetric with respect to the real line [2].”