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$.
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ISSN = {1609-3321},
MRCLASS = {32S22 (17B81 81R12 82C23)},
MRNUMBER = {2007m:32016},
MRREVIEWER = {V. A. Golubeva},
ZBLNUMBER = {05184506},
} -
[Ve]
J. Verschelde, "Numerical evidence for a conjecture in real algebraic geometry," Experiment. Math., vol. 9, iss. 2, pp. 183-196, 2000.
@article{Ve, MRKEY = {1780205},
AUTHOR = {Verschelde, Jan},
TITLE = {Numerical evidence for a conjecture in real algebraic geometry},
JOURNAL = {Experiment. Math.},
FJOURNAL = {Experimental Mathematics},
VOLUME = {9},
YEAR = {2000},
NUMBER = {2},
PAGES = {183--196},
ISSN = {1058-6458},
MRCLASS = {65H10 (14M15 65H20)},
MRNUMBER = {2001i:65062},
URL = {http://projecteuclid.org/getRecord?id=euclid.em/1045952344},
ZBLNUMBER = {1054.14080},
MRREVIEWER = {Shuguang Wang},
}