The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

• 成果类型: Article
• 署名作者: Mukhin, Evgeny; Tarasov, Vitaly; Varchenko, Alexander
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.863
• 发表日期: 2009
• 页码: 863-881
• 关键词: rational functions; schubert calculus; critical-points; arrangements; hyperplanes

摘要

We prove the B. and M. Shapiro conjecture that if the Wronskian of a set of polynomials has real roots only, then the complex span of this set of polynomials has a basis consisting of polynomials with real coefficients. This, in particular, implies the following result: If all ramification points of a parametrized rational curve phi : CP1 -> CPr lie on a circle in the Riemann sphere CP1, then phi maps this circle into a suitable real subspace RPr subset of CPr. The proof is based on the Bethe ansatz method in the Gaudin model. The key observation is that a symmetric linear operator on a Euclidean space has real spectrum. In Appendix A, we discuss properties of differential operators associated with Bethe vectors in the Gaudin model. In particular, we prove a statement, which may be useful in complex algebraic geometry; it claims that certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of the corresponding Gaudin Hamiltonians is simple. In Appendix B, we formulate a conjecture on reality of orbits of critical points of master functions and prove this conjecture for master functions associated with Lie algebras of types A(r) B-r and C-r.