## The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz.(English)Zbl 1213.14101

Let $$V$$ be an $$r+1$$ dimensional complex linear subspace of $$\mathbb{C}[x]$$. The Wronskian of $$V$$ is defined as the Wronskian of an ordered basis of $$V$$; hence it is defined up to multiplication by a non-zero scalar. The main result of the paper is the proof of the following theorem (earlier B. and M. Shapiro conjecture): If the roots of the Wronskian $$Wr$$ of $$V$$ are real, then $$V\subset \mathbb{R}[x] \subset \mathbb{C}[x]$$. A continuity argument shows that it is enough to prove this theorem in the case when $$Wr$$ has simple roots. The authors recall a Schubert calculus argument to find an upper bound for the number of subspaces $$V \subset C[x]$$ with a given exponent at infinity (degrees of a basis) and $$Wr=\prod (x-z_i)$$, where the $$z_i$$’s are distinct. In Sections 2 and 3 the Bethe ansatz construction for a Gaudin model is studied and used to construct the expected number of subspaces with given exponent and given $$Wr$$. A thorough study of the Gaudin Hamiltonians shows that the constructed subspaces $$V$$ are all real if the roots of $$Wr$$ are all real. This proves the main theorem.
The authors also discuss some related topics. They show that their main theorem implies that if the ramified points of a map $$\phi: \mathbb{C}\mathbb{P}^1 \to \mathbb{C}\mathbb{P}^r$$ are contained in a circle in $$\mathbb{C}\mathbb{P}^1$$, then $$\phi$$ of this circle is contained in an (appropriate) $$\mathbb{R}\mathbb{P}^r \subset \mathbb{C}\mathbb{P}^r$$. They also prove a statement about how the transversality of intersections of Schubert cycles in Grassmannians is related to the simplicity of the spectrum of Gaudin Hamiltonians. In Appendix B they show that their results imply the reality of orbits of critical points of master functions for the root systems A, B, and C; and conjecture this statement in general.

### MSC:

 14N15 Classical problems, Schubert calculus 17B80 Applications of Lie algebras and superalgebras to integrable systems 34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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### References:

 [1] A. V. Chervov and D. Talalaev, Universal $$G$$-oper and Gaudin eigenproblem, preprint, 2004. [2] D. Eisenbud and J. Harris, ”Divisors on general curves and cuspidal rational curves,” Invent. Math., vol. 74, iss. 3, pp. 371-418, 1983. · Zbl 0527.14022 [3] T. Ekedahl, B. Shapiro, and M. Shapiro, ”First steps towards total reality of meromorphic functions,” Mosc. Math. J., vol. 6, iss. 1, pp. 95-106, 222, 2006. · Zbl 1126.14064 [4] A. Eremenko and A. Gabrielov, ”Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry,” Ann. of Math., vol. 155, iss. 1, pp. 105-129, 2002. · Zbl 0997.14015 [5] A. Eremenko and A. Gabrielov, ”Degrees of real Wronski maps,” Discrete Comput. Geom., vol. 28, iss. 3, pp. 331-347, 2002. · Zbl 1004.14011 [6] A. Eremenko and A. Gabrielov, ”Elementary proof of the B. and M. Shapiro conjecture for rational functions,” , preprint , 2005. · Zbl 1246.14062 [7] A. Eremenko, A. Gabrielov, M. Shapiro, and A. Vainshtein, ”Rational functions and real Schubert calculus,” Proc. Amer. Math. Soc., vol. 134, iss. 4, pp. 949-957, 2006. · Zbl 1110.14052 [8] V. G. Kac, Infinite-Dimensional Lie Algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990. · Zbl 0716.17022 [9] V. Kharlamov and F. Sottile, ”Maximally inflected real rational curves,” Mosc. Math. J., vol. 3, iss. 3, pp. 947-987, 1199, 2003. · Zbl 1052.14070 [10] P. P. Kulish and E. K. Sklyanin, ”Quantum spectral transform method. Recent developments,” in Integrable Quantum Field Theories, Hietarinta, J. and Montonen, C., Eds., New York: Springer-Verlag, 1982, vol. 151, pp. 61-119. · Zbl 0734.35071 [11] A. Matsuo, ”An application of Aomoto-Gel\cprime fand hypergeometric functions to the $${ SU}(n)$$ Knizhnik-Zamolodchikov equation,” Comm. Math. Phys., vol. 134, iss. 1, pp. 65-77, 1990. · Zbl 0714.33012 [12] A. Molev, M. Nazarov, and G. Olcprimeshanskiui, ”Yangians and classical Lie algebras,” Uspekhi Mat. Nauk, vol. 51, iss. 2(308), pp. 27-104, 1996. · Zbl 0876.17014 [13] E. Mukhin, V. Tarasov, and A. Varchenko, ”Bethe eigenvectors of higher transfer matrices,” J. Stat. Mech. Theory Exp., vol. 8, p. 08002, 2006. [14] E. Mukhin and A. Varchenko, ”Remarks on critical points of phase functions and norms of Bethe vectors,” in Arrangements, Falk, M. and Terao, H., Eds., Tokyo: Kinokuniya, 2000, pp. 239-246. · Zbl 1040.17001 [15] E. Mukhin and A. Varchenko, ”Critical points of master functions and flag varieties,” Commun. Contemp. Math., vol. 6, iss. 1, pp. 111-163, 2004. · Zbl 1050.17022 [16] E. Mukhin and A. Varchenko, ”Norm of a Bethe vector and the Hessian of the master function,” Compos. Math., vol. 141, iss. 4, pp. 1012-1028, 2005. · Zbl 1072.82012 [17] R. Rimányi, L. Stevens, and A. Varchenko, ”Combinatorics of rational functions and Poincaré-Birchoff-Witt expansions of the canonical $$U(\mathfrak n_-)$$-valued differential form,” Ann. Comb., vol. 9, iss. 1, pp. 57-74, 2005. · Zbl 1088.33007 [18] J. Ruffo, Y. Sivan, E. Soprunova, and F. Sottile, ”Experimentation and conjectures in the real Schubert calculus for flag manifolds,” Experiment. Math., vol. 15, iss. 2, pp. 199-221, 2006. · Zbl 1111.14049 [19] N. Reshetikhin and A. Varchenko, ”Quasiclassical asymptotics of solutions to the KZ equations,” in Geometry, Topology, & Physics, Yau, S. -T., Ed., Cambridge, MA: Internat. Press, 1995, pp. 293-322. · Zbl 0867.58065 [20] V. V. Schechtman and A. N. Varchenko, ”Arrangements of hyperplanes and Lie algebra homology,” Invent. Math., vol. 106, iss. 1, pp. 139-194, 1991. · Zbl 0754.17024 [21] I. Scherbak and A. Varchenko, ”Critical points of functions, $$\mathfrak{sl}_2$$ representations, and Fuchsian differential equations with only univalued solutions,” Mosc. Math. J., vol. 3, iss. 2, pp. 621-645, 745, 2003. · Zbl 1039.34077 [22] F. Sottile, ”\relax Enumerative geometry for the real Grassmannian of lines in projective space,” Duke Math. J., vol. 87, iss. 1, pp. 59-85, 1997. · Zbl 0986.14033 [23] F. Sottile, ”Enumerative geometry for real varieties,” in Algebraic Geometry, Kollár, J., Lazarsfeld, R., and Morrison, D. R., Eds., Providence, RI: Amer. Math. Soc., 1997, pp. 435-447. · Zbl 0986.14033 [24] F. Sottile, ”The special Schubert calculus is real,” Electron. Res. Announc. Amer. Math. Soc., vol. 5, pp. 35-39, 1999. · Zbl 0921.14037 [25] F. Sottile, ”Real Schubert calculus: Polynomial systems and a conjecture of Shapiro and Shapiro,” Experiment. Math., vol. 9, iss. 2, pp. 161-182, 2000. · Zbl 0997.14016 [26] F. Sottile, ”Enumerative real algebraic geometry,” in Algorithmic and Quantitative Real Algebraic Geometry, Basu, S. and Gonzalez-Vega, L., Eds., Providence, RI: Amer. Math. Soc., 2003, pp. 139-179. · Zbl 1081.14080 [27] F. Sottile, ”The conjecture of Shapiro and Shapiro,” Experiment. Math., web page , 2000. · Zbl 0997.14016 [28] D. Talalaev, ”Quantization of the Gaudin system,” , preprint , 2004. · Zbl 1111.82015 [29] A. Varchenko, Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups, River Edge, NJ: World Sci. Publ., 1995. · Zbl 0951.33001 [30] A. Varchenko, ”Bethe ansatz for arrangements of hyperplanes and the Gaudin model,” Moscow Math. Jour., vol. 6, iss. 1, pp. 195-210, 223, 2006. · Zbl 1375.32050 [31] J. Verschelde, ”Numerical evidence for a conjecture in real algebraic geometry,” Experiment. Math., vol. 9, iss. 2, pp. 183-196, 2000. · Zbl 1054.14080
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