The Experts below are selected from a list of 210 Experts worldwide ranked by ideXlab platform
Alexander Varchenko - One of the best experts on this subject based on the ideXlab platform.
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
Annals of Mathematics, 2009Co-Authors: E. Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract: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 .
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
arXiv: Algebraic Geometry, 2005Co-Authors: E. Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract: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 $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset 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.
E. Mukhin - One of the best experts on this subject based on the ideXlab platform.
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
Annals of Mathematics, 2009Co-Authors: E. Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract: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 .
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
arXiv: Algebraic Geometry, 2005Co-Authors: E. Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract: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 $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset 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.
Vitaly Tarasov - One of the best experts on this subject based on the ideXlab platform.
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
Annals of Mathematics, 2009Co-Authors: E. Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract: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 .
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The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz
arXiv: Algebraic Geometry, 2005Co-Authors: E. Mukhin, Vitaly Tarasov, Alexander VarchenkoAbstract: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 $ f : CP^1 \to CP^r $ lie on a circle in the Riemann sphere $ CP^1 $, then $f$ maps this circle into a suitable real subspace $ RP^r \subset 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.
Sticlaru Gabriel - One of the best experts on this subject based on the ideXlab platform.
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Freeness and invariants of Rational plane Curves
American Mathematical Society, 2020Co-Authors: Busé Laurent, Dimca Alexandru, Sticlaru GabrielAbstract:International audienceGiven a parameterization φ of a Rational plane Curve C, we study some invariants of C via φ. We first focus on the characterization of Rational cuspidal Curves, in particular we establish a relation between the discriminant of the pull-back of a line via φ, the dual Curve of C and its singular points. Then, by analyzing the pull-backs of the global differential forms via φ, we prove that the (nearly) freeness of a Rational Curve can be tested by inspecting the Hilbert function of the kernel of a canonical map. As a by product, we also show that the global Tjurina number of a Rational Curve can be computed directly from one of its parameterization, without relying on the computation of an equation of C
Thomas W Sederberg - One of the best experts on this subject based on the ideXlab platform.
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A new implicit representation of a planar Rational Curve with high order singularity
Computer Aided Geometric Design, 2002Co-Authors: Falai Chen, Thomas W SederbergAbstract:This paper presents a new representation for the implicit equation of a planar Rational Curve with high order singularity by using moving Curves technique. The new representation is the determinant of a matrix which is less than one half the size of the conventional expression based on the Bezout's resultant. An efficient algorithm to compute the new representation is derived, and comparisons for the computational costs between various resultant based methods are made. The results show that the implicit representation developed in this paper is not only more compact but also much more efficient to compute the implicit equation of the Rational Curve than previous methods.
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the mu basis of a Rational ruled surface
Computer Aided Geometric Design, 2001Co-Authors: Falai Chen, Jianmin Zheng, Thomas W SederbergAbstract:Abstract The mu-basis of a planar Rational Curve is a polynomial ideal basis comprised of two polynomials that greatly facilitates computing the implicit equation of the Curve. This paper defines a mu-basis for a Rational ruled surface, and presents a simple algorithm for computing the mu-basis. The mu-basis consists of two polynomials p(x,y,z,s) and q(x,y,z,s) that are linear in x,y,z and degree μ and m−μ in s respectively, where m is the degree of the implicit equation. The implicit equation of the surface is then obtained by merely taking the resultant of p and q with respect to s . This implicitization algorithm is faster and/or more robust than previous methods.
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on the minors of the implicitization bezout matrix for a Rational plane Curve
Computer Aided Geometric Design, 2001Co-Authors: Engwee Chionh, Thomas W SederbergAbstract:Abstract This paper investigates the first minors M i,j of the Bezout matrix used to implicitize a degree- n plane Rational Curve P (t) . It is shown that the degree n−1 Curve M i,j =0 passes through all of the singular points of P (t) . Furthermore, the only additional points at which M i,j =0 and P (t) intersect are an (i+j) -fold intersection at P (0) and a (2n−2−i−j) -fold intersection at P (∞) . Thus, a polynomial whose roots are exactly the parameter values of the singular points of P (t) can be obtained by intersecting P (t) with M 0,0 . Previous algorithms of finding such a polynomial are less direct. We further show that M i,j =M k,l if i+j=k+l . The method also clarifies the applicability of inversion formulas and yields simple checks for the existence of singularities in a cubic Bezier Curve.
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implicitizing Rational Curves by the method of moving algebraic Curves
Journal of Symbolic Computation, 1997Co-Authors: Thomas W Sederberg, Ron Goldman, Hang DuAbstract:Abstract A function F(x,y,t) that assigns to each parameter t an algebraic Curve F(x,y,t)=0 is called a moving Curve. A moving Curve F(x,y,t) is said to follow a Rational Curve x=x(t)/w(t) , y=y(t)/w(t) if F(x(t)/w(t), y(t)/w(t),t) is identically zero. A new technique for finding the implicit equation of a Rational Curve based on the notion of moving conics that follow the Curve is investigated. For Rational Curves of degree 2 n with no base points the method of moving conics generates the implicit equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y , whereas standard resultant methods generate the implicit equation as the determinant of a 2 n × 2 n matrix where each entry is a linear polynomial in x and y . Thus implicitization using moving conics yields more compact representations for the implicit equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.
