Degrees of Real Wronski Maps

Eremenko,; Gabrielov,

doi:10.1007/s00454-002-0735-x

Cited by 43 publications

(75 citation statements)

References 18 publications

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“…The conjecture is proved in [EG02b] for r D 1; see a more elementary proof also for r D 1 in [EG05]. The conjecture, its supporting evidence, and applications are discussed in [EG02b], [EG02a], [EG05], [EGSV06], [ESS06], [KS03], [RSSS06], [Sot97a], [Sot97b], [Sot99], [Sot00b], [Sot03], [Sot00a] and [Ver00].…”

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confidence: 99%

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

2009

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We prove the B. and M. Shapiro conjecture: 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: certain Schubert cycles in a Grassmannian intersect transversally if the spectrum of suitable 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 , C r .

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confidence: 99%

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

2009

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“…The most promising approach to this conjecture seems to be the study of the geometry of the Wronski map from Grassmannians to projective spaces, see e.g. [EG2]. Namely, consider the map W k,n : G k+1,n+1 → RP (k+1)(n−k) associating to a (k + 1)-dimensional linear subspace in the space of polynomials its Wronskian, i.e.…”

Section: Explanations Tomentioning

confidence: 99%

On Two Conjectures Concerning Convex Curves

Sedykh

Shapiro

2005

Int. J. Math.

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Abstract. In this paper we recall two basic conjectures on the developables of convex projective curves, prove one of them and disprove the other in the first nontrivial case of curves in RP 3 . Namely, we show that i) the tangent developable of any convex curve in RP 3 has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 3 such that no real line intersects all four of them. The question (discussed in [EG1] and [So4]) whether the second conjecture is true in the special case of rational normal curves still remains open. §1. Introduction and resultsWe start with some important notions.Main definition. A smooth closed curve γ : S 1 → RP n is called locally convex if the local multiplicity of intersection of γ with any hyperplane H ⊂ RP n at any of the intersection points does not exceed n = dim RP n and globally convex or just convex if the above condition holds for the global multiplicity, i.e for the sum of local multiplicities.Local convexity of γ is a simple requirement of nondegeneracy of the osculating Frenet n-frame of γ, i.e. of the linear independence of γ ′ (t), ..., γ (n) (t) for any t ∈ S 1 . Global convexity is a nontrivial property equivalent to the fact that the (n + 1)-tuple of γ's homogeneous coordinates forms a Tschebychev system of functions, see e.g. [KS]. The simplest examples of convex curves are the rational normal curve ρ n : t → (t, t 2 , . . . , t n ) (in some affine coordinates on RP n ) and the standard trigonometric curve τ 2k : t → (sin t, cos t, sin 2t, cos 2t, . . . , sin 2kt, cos 2kt) (in some affine coordinates on RP 2k ).Definition. The k-th developable D k (γ) of a curve γ : S 1 → RP n is the union of all k-dimensional osculating subspaces to γ. The hypersurfaceNote that D k (γ) can be considered as the image of the natural associated map γ k :Definition. Let M be a compact manifold of some dimension l ≤ n and φ : M → RP n be a smooth map. By the degree of φ(M ) we understand the supremum of the number of its intersection points withV. SEDYKH AND B. SHAPIRO subspaces for φ(M ) (for example, if dim φ(M ) < dim(M )) then we set the degree of φ(M ) equal to zero. (It is rather obvious that if the Jacobian of φ is nondegenerate at least at one point of M then generic (n − l)-dimensional subspaces exist.)In particular, one can consider the degree of D k (γ) which is positive unless dim D k (γ) < k + 1.

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“…Eremenko and Gabrielov [3] showed that the number of real solutions to certain systems of real polynomial equations is bounded below by the topological degree of the corresponding Wronski map, and computed this degree. Soprunova and Sottile [19] extended the definition of the real Wronski map to certain sparse polynomial systems associated with partially ordered sets, computed its degree and derived the corresponding lower bounds on the number of real solutions.…”

Section: Introductionmentioning

confidence: 99%

Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

Azar

Gabrielov

2010

Discrete Comput Geom

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Abstract. The B. and M. Shapiro conjecture stated that all solutions of the Schubert Calculus problems associated with real points on the rational normal curve should be real. For Grassmannians, it was proved by Mukhin, Tarasov and Varchenko. For flag varieties, Sottile found a counterexample and suggested that all solutions should be real under certain monotonicity conditions. In this paper, we compute lower bounds on the number of real solutions for some special cases of the B. and M. Shapiro conjecture for flag varieties, when Sottile's monotonicity conditions are not satisfied.

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Degrees of Real Wronski Maps

Cited by 43 publications

References 18 publications

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

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

On Two Conjectures Concerning Convex Curves

Some Lower Bounds in the B. and M. Shapiro Conjecture for Flag Varieties

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