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

By E. Mukhin, V. Tarasov, A. Varchenko. 2005

117 citations recorded by the source corpus.

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 φ: ℂℙ 1 → ℂℙ r lie on a circle in the Riemann sphere ℂℙ 1 , then φ maps this circle into a suitable real subspace ℝℙ r ⊂ 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 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 .

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c7fccc36ed2b7fb98765657ef98aad109b52f2c6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

1. V. Tarasov
2. E. Mukhin
3. A. Varchenko
4. Alexandre Eremenko
5. Andrei Gabrielov

In Geometry and Topology

1. V. Tarasov
2. A. Varchenko
3. E. Mukhin
4. Alexandre Eremenko
5. Andrei Gabrielov
6. Valerio Toledano Laredo
7. P. Boalch

In Statistical and Nonlinear Physics

1. V. Tarasov
2. A. Varchenko
3. E. Mukhin
4. Valerio Toledano Laredo
5. P. Boalch

Citations in the corpus

• Metrics with conic singularities and spherical polygons (2014, background): “The problem is equivalent to classification of some second order linear differential equations with regular singularities, with real parameters and unitary monodromy.”
• Bispectral and (gl_N, gl_M) Dualities, Discrete Versus Differential (2006, background): “ the intersection in (7.1) is equal to the multiplicity of the trivial slN-module in the tensor product of slN-modules Le hNi c(1) ⊗ ...⊗ Le hNi c(M) ⊗ Le c(∞1) ⊗ LehNi c(∞2) (7.2) . Proposition 7.3 ([MTV2]). The multiplicity of the trivial slN-module in the tensor product (7.2) is equal to the dime”
• Schubert calculus and representations of the general linear group (2007, methodology): “that the action of the polynomial PN i=0 C¯ i(bs) QN−i−1 j=0 (α − j) on M λ,a ⊂ ⊗ k r=1L λ(r)(br) coincides with the action of the operator Yk r=1 r6= s (bs −br) nr 1⊗(s−1) ⊗ Z(α− N +1)⊗1⊗(k−s) , cf. [MTV2]. The lemma follows from Theorem 2.1 and formula (2.3). Hence, the projection B λ,a → BΛ,λ,b d”