Rational functions with real critical points and the B. and M.!Shapiro conjecture in real enumerative geometry

• 成果类型: Article
• 署名作者: Eremenko, A; Gabrielov, A
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3062151
• 发表日期: 2002
• 页码: 105-129
• 关键词: schubert calculus

摘要

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.