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.

Latex, 18 pages, revised version