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

Mukhin, E.; Tarasov, V.; Varchenko, A.

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

Authors

Mukhin, E.

Tarasov, V.

Varchenko, A.

Description

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 : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset CP^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