{
  "id": "math/0512299",
  "version": "v2",
  "published": "2005-12-14T01:36:14.000Z",
  "updated": "2006-05-02T18:58:33.000Z",
  "title": "The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz",
  "authors": [
    "E. Mukhin",
    "V. Tarasov",
    "A. Varchenko"
  ],
  "comment": "Latex, 18 pages, revised version",
  "categories": [
    "math.AG",
    "math.QA"
  ],
  "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 : 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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-05-02T18:58:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "real algebraic geometry",
      "shapiro conjecture",
      "gaudin model",
      "polynomials",
      "symmetric linear operator"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12299M"
    }
  }
}