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25
Rational functions with real critical points and the B. and M. Shapiro conjecture in real enumerative geometry
, 2001
"... Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)inddimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the dth Catalan n ..."
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Cited by 57 (16 self)
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Suppose that 2d − 2 tangent lines to the rational normal curve z ↦ → (1: z:...: z d)inddimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the dth Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann sphere (for example, on the real line), then the function maps this circle into a circle.
Rational functions and real Schubert calculus
 Proc. AMS (electronically published on
, 2005
"... Abstract. We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable. 1. ..."
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Cited by 23 (7 self)
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Abstract. We single out some problems of Schubert calculus of subspaces of codimension 2 that have the property that all their solutions are real whenever the data are real. Our arguments explore the connection between subspaces of codimension 2 and rational functions of one variable. 1.
Numerical Evidence For A Conjecture In Real Algebraic Geometry
, 1998
"... Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are over ..."
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Cited by 22 (4 self)
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Homotopies for polynomial systems provide computational evidence for a challenging instance of a conjecture about whether all solutions are real. The implementation of SAGBI homotopies involves polyhedral continuation, flat deformation and cheater's homotopy. The numerical difficulties are overcome if we work in the true synthetic spirit of the Schubert calculus by selecting the numerically most favorable equations to represent the geometric problem. Since a wellconditioned polynomial system allows perturbations on the input data without destroying the reality of the solutions we obtain not just one instance, but a whole manifold of systems that satisfy the conjecture. Also an instance that involves totally positive matrices has been verified. The optimality of the solving procedure is a promising first step towards the development of numerically stable algorithms for the pole placement problem in linear systems theory.
Pole Placement by Static Output Feedback for Generic Linear Systems
 in the Introduction
, 2001
"... We consider linear systems with m inputs, p outputs and McMillan degree n, such that n = mp. If both m and p are even, we show that there is a nonempty open (in the usual topology) subset U of such systems, where the real pole placement map is not surjective. It follows that for each system in U , ..."
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Cited by 17 (5 self)
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We consider linear systems with m inputs, p outputs and McMillan degree n, such that n = mp. If both m and p are even, we show that there is a nonempty open (in the usual topology) subset U of such systems, where the real pole placement map is not surjective. It follows that for each system in U , there exists an open set of pole configurations, symmetric with respect to the real line, which cannot be assigned by any real static output feedback. 1.
Jeu de taquin and a monodromy problem for Wronskians of polynomials
, 2009
"... The Wronskian associates to d linearly independent polynomials of degree at most n, a nonzero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy grou ..."
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Cited by 8 (1 self)
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The Wronskian associates to d linearly independent polynomials of degree at most n, a nonzero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schützenberger’s jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the LittlewoodRichardson rule.
First step towards total reality of meromorphic functions
 Mosc. Math. J. vol
, 2006
"... Abstract. It was earlier conjectured by the second and the third authors that any rational curve γ: CP 1 → CP n such that the inverse images of all its flattening points lie on the real line RP 1 ⊂ CP 1 is real algebraic up to a linear fractional transformation of the image CP n, see [1], [8] and [1 ..."
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Cited by 7 (1 self)
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Abstract. It was earlier conjectured by the second and the third authors that any rational curve γ: CP 1 → CP n such that the inverse images of all its flattening points lie on the real line RP 1 ⊂ CP 1 is real algebraic up to a linear fractional transformation of the image CP n, see [1], [8] and [12]. (By a flattening point p on γ we mean a point at which the Frenet nframe (γ ′ , γ ′ ′ ,..., γ (n)) is degenerate.) Below we extend this conjecture to the case of meromorphic functions on real algebraic curves of higher genera and settle it for meromorphic functions of degrees 2,3 and several other cases. To Victor Vassiliev on the occasion of his fiftieth birthday 1.
From enumerative geometry to solving systems of polynomials equations, Computations in algebraic geometry with Macaulay 2
 Algorithms Comput. Math
, 2002
"... Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerat ..."
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Cited by 7 (1 self)
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Solving a system of polynomial equations is a ubiquitous problem in the applications of mathematics. Until recently, it has been hopeless to find explicit solutions to such systems, and mathematics has instead developed deep and powerful theories about the solutions to polynomial equations. Enumerative Geometry is concerned with counting the number of solutions when the polynomials come from a geometric situation and Intersection Theory gives methods to accomplish the enumeration. We use Macaulay 2 to investigate some problems from enumerative geometry, illustrating some applications of symbolic computation to this important problem of solving systems of polynomial equations. Besides enumerating solutions to the resulting polynomial systems, which include overdetermined, deficient, and improper systems, we address the important question of real solutions to these geometric problems. 1
On two conjectures concerning convex curves
 Internat. J. Math. vol
"... 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 ..."
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Cited by 6 (0 self)
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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. We start with some important notions. §1. Introduction and results 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.
THE B. AND M. SHAPIRO CONJECTURE IN REAL ALGEBRAIC GEOMETRY AND THE BETHE ANSATZ
, 2005
"... 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 rami ..."
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Cited by 4 (0 self)
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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 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 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. 1. The B. and M. Shapiro conjecture 1.1. Statement of the result. Fix a natural number r � 1. Let V ⊂ C[x] be a vector subspace of dimension r + 1. The space V is called real if it has a basis consisting of polynomials in R[x]. For a given V, there exists a unique linear differential operator D = dr+1
Two Conjectures on Convex Curves
, 2002
"... 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 . Namely, we show i) that the tangent developable of any convex curve in has degree 4 and ii) construct an example of 4 tange ..."
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Cited by 4 (0 self)
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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 . Namely, we show i) that the tangent developable of any convex curve in has degree 4 and ii) construct an example of 4 tangent lines to a convex curve in RP 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.