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

Abstract

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 $\phi:\Bbb{C}\mathbb P^1 \to \Bbb{C}\mathbb P^r$ lie on a circle in the Riemann sphere $\Bbb{C}\mathbb P^1$, then $\phi$ maps this circle into a suitable real subspace $\mathbb R\mathbb P^r \subset \Bbb{C}\mathbb P^r$.

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 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$.