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

Simple item page
dc.creatorMukhin, E.
dc.creatorTarasov, V.
dc.creatorVarchenko, A.
dc.date2005-12-14
dc.date2006-05-02
dc.date.accessioned2026-07-07T06:55:12Z
dc.date.available2026-07-07T06:55:12Z
dc.descriptionWe 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.
dc.descriptionLatex, 18 pages, revised version
dc.identifierhttps://arxiv.org/abs/math/0512299
dc.identifierhttp://arxiv.org/abs/math/0512299
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/106207
dc.subjectAlgebraic Geometry
dc.subjectQuantum Algebra
dc.titleThe B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
dc.typetext

Files

Collections

arXiv