Results 1  10
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106
A Gröbner free alternative for polynomial system solving
 Journal of Complexity
, 2001
"... Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic ..."
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Cited by 82 (16 self)
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Given a system of polynomial equations and inequations with coefficients in the field of rational numbers, we show how to compute a geometric resolution of the set of common roots of the system over the field of complex numbers. A geometric resolution consists of a primitive element of the algebraic extension defined by the set of roots, its minimal polynomial and the parametrizations of the coordinates. Such a representation of the solutions has a long history which goes back to Leopold Kronecker and has been revisited many times in computer algebra. We introduce a new generation of probabilistic algorithms where all the computations use only univariate or bivariate polynomials. We give a new codification of the set of solutions of a positive dimensional algebraic variety relying on a new global version of Newton’s iterator. Roughly speaking the complexity of our algorithm is polynomial in some kind of degree of the system, in its height, and linear in the complexity of evaluation
Generalized Bezout Identity
, 1998
"... We describe a new approach of the generalized Bezout identity for linear timevarying ordinary differential control systems. We also explain when and how it can be extended to linear partial differential control systems. We show that it only depends on the algebraic nature of the differential module ..."
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Cited by 18 (10 self)
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We describe a new approach of the generalized Bezout identity for linear timevarying ordinary differential control systems. We also explain when and how it can be extended to linear partial differential control systems. We show that it only depends on the algebraic nature of the differential module determined by the equations of the system. This formulation shows that the generalized Bezout identity is equivalent to the splitting of an exact differential sequence formed by the control system and its parametrization. This point of view gives a new algebraic and geometric interpretation of the entries of the generalized Bezout identity.
Equivalences of Linear Control Systems
, 2000
"... We show how homological algebra and algebraic analysis allow to give various notions of equivalence for linear control systems which do not depend on their presentations and therefore preserve their structural properties. ..."
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Cited by 15 (9 self)
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We show how homological algebra and algebraic analysis allow to give various notions of equivalence for linear control systems which do not depend on their presentations and therefore preserve their structural properties.
The quantum Euler class and the quantum cohomology of the Grassmannians
"... The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element; ” in the classical case this is the Euler class, and in the quantum case ..."
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Cited by 12 (1 self)
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The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element; ” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class. ” We prove that the characteristic element of a Frobenius algebra A is a unit if and only if A is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] LandauGinzbug potential. 1
On Codes, Matroids and Secure MultiParty Computation from Linear Secret Sharing Schemes
 In Proceedings of CRYPTO 2005, volume 3621 of LNCS
, 2004
"... Error correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, we study the connections between codes, matroids and a special class of secret sharing schemes, namely multiplicative linear secret sharing schemes (LSSSs). Such schemes are k ..."
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Cited by 12 (7 self)
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Error correcting codes and matroids have been widely used in the study of ordinary secret sharing schemes. In this paper, we study the connections between codes, matroids and a special class of secret sharing schemes, namely multiplicative linear secret sharing schemes (LSSSs). Such schemes are known to enable multiparty computation protocols secure against general (nonthreshold) adversaries.
Casimir Operators and Monodromy Representations of Generalised Braid Groups
, 2003
"... Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group ..."
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Cited by 10 (6 self)
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Let g be a complex, simple Lie algebra with Cartan subalgebra h and Weyl group W. We construct a one–parameter family of flat connections ∇κ on h with values in any finite–dimensional g–module V and simple poles on the root hyperplanes. The corresponding monodromy representation of the braid group Bg of type g is a deformation of the action of (a finite extension of) W on V. The residues of ∇κ are the Casimirs κα of the subalgebras sl α 2 ⊂ g corresponding to the roots of g. The irreducibility of a subspace U ⊆ V under the κα implies that, for generic values of the parameter, the braid group Bg acts irreducibly on U. Answering a question of Knutson and Procesi, we show that these Casimirs act irreducibly on the weight spaces of all simple g–modules if g = sl3 but that this is not the case if g ≇ sl2, sl3. We use this to disprove a conjecture of Kwon and Lusztig stating the irreducibility of quantum Weyl group actions of Artin’s braid group Bn on the zero weight spaces of all simple U�sln–modules for n ≥ 4. Finally, we study the irreducibility of the action of the Casimirs on the zero weight spaces of self–dual g–modules and obtain complete classification results for g = sln and g2 and conjecturally complete results for g orthogonal or symplectic.
Some Generic Results on Algebraic Observability and Connections with Realization Theory
 Proc. 2nd European Control Conf
, 1993
"... We analyze Glad/Fliess algebraic observability for polynomial control systems from a commutative algebraic/algebrogeometric point of view, using some results from the theory of Gröbner bases. Furthermore, we discuss some topics in realization theory for polynomial differential equations. Most issue ..."
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Cited by 9 (6 self)
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We analyze Glad/Fliess algebraic observability for polynomial control systems from a commutative algebraic/algebrogeometric point of view, using some results from the theory of Gröbner bases. Furthermore, we discuss some topics in realization theory for polynomial differential equations. Most issues are treated in a constructive framework.
Linear Equivalence of Ideal Topologies
"... It is proved that whenever P is a prime ideal in a commutative Noetherian ring such that the Padic and the Psymbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to nontorsion ..."
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Cited by 9 (0 self)
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It is proved that whenever P is a prime ideal in a commutative Noetherian ring such that the Padic and the Psymbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to nontorsion points on nonsingular elliptic cubic curves.
Rings of differential operators on classical rings of invariants
 MEMOIRS OF THE AMS
, 1989
"... We consider rings of differential operators over the classical rings of invariants, in the sense of Weyl [We]. Thus, let X k be one of the following varieties: (CASE A) all complex p × q matrices of rank ≤ k; (CASE B) all symmetric n × n matrices of rank ≤ k; (CASE C) all antisymmetric n × n matrice ..."
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Cited by 9 (0 self)
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We consider rings of differential operators over the classical rings of invariants, in the sense of Weyl [We]. Thus, let X k be one of the following varieties: (CASE A) all complex p × q matrices of rank ≤ k; (CASE B) all symmetric n × n matrices of rank ≤ k; (CASE C) all antisymmetric n × n matrices of rank ≤ 2k. We prove that the ring of differential operators D(X k) = D(O(X k)) defined on the ring of regular functions O(X k) is a simple, finitely generated, Noetherian domain. Assume further that X k is singular (which is the only interesting case). Then the result is proved by showing that D(X k) is a factor ring of an enveloping algebra U(g). Here g = gl(p + q) , sp(2n) and so(2n) in the Cases A, B and C, respectively. Finally, let SO(k) act in the natural way on the ring C[X] of complex polynomials in kn variables. Then we prove that D(C[X] SO(k) ) has a similarly pleasant structure and, at least for k ≤ n, is a finitely generated U(sp(2n))module.