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

摘要: We prove the B. and M. Shapiro conjecture that if Wronskian of a set polynomials has real roots only, then complex span this basis consisting with coefficients. This, in particular, implies following result: If all ramification points parametrized rational curve φ: ℂℙ 1 → r lie on circle Riemann sphere , φ maps into suitable subspace ℝℙ ⊂ . The proof is based Bethe ansatz method Gaudin model. key observation symmetric linear operator Euclidean space spectrum. In Appendix A, we discuss properties differential operators associated vectors statement, which may be useful algebraic geometry; it claims certain Schubert cycles Grassmannian intersect transversally spectrum corresponding Hamiltonians simple. B, formulate reality orbits critical master functions for Lie algebras types A B C