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

Abstract

We prove the B. and M. Shapiro conjecture that says 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 $f:C{P}^1\to C{P}^r$ lie on a circle in the Riemann sphere $C{P}^1$, then $f$ maps this circle into a suitable real subspace $R{P}^r\subset C{P}^r$. 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 a real spectrum. In Appendix we discuss properties of differential operators associated with Bethe vectors in the Gaudin model and, in particular, prove a conditional statement: we deduce the transversality of certain Schubert cycles in a Grassmannian from the simplicity of the spectrum of the Gaudin Hamiltonians.

arXiv commentLatex, 18 pages, revised version

References in corpus (4)

• Quasiclassical asymptotics of solutions to the KZ equations
• Quantization of the Gaudin System
• Universal G-oper and Gaudin eigenproblem
• Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model

Cited by in corpus (10)

• On the Bethe Ansatz for the Jaynes-Cummings-Gaudin model
• Limits of Gaudin algebras, quantization of bending flows, Jucys--Murphy elements and Gelfand--Tsetlin bases
• Schubert calculus and representations of general linear group
• Jeu de taquin and a monodromy problem for Wronskians of polynomials
• Decomplexification of the Capelli identities and holomorphic factorization
• Differential equation for Jacobi-Pineiro polynomials
• Reality and transversality for Schubert calculus in OG(n,2n+1)
• Higher Lame Equations and Critical Points of Master Functions
• Towards the generalized Shapiro and Shapiro conjecture
• General isotropic flags are general (for Grassmannian Schubert calculus)