Abstract not found

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

We describe a new approach of the generalized Bezout identity for linear time-varying 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.

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.

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 (non-threshold) adversaries.

We analyze Glad/Fliess algebraic observability for polynomial control systems from a commutative algebraic/algebro-geometric 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.

Abstract. 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. Contents

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.

It is known that the underlying spaces of all abelian quotient singularities which are embeddable as complete intersections of hypersurfaces in an affine space can be overall resolved by means of projective torus-equivariant crepant birational morphisms in all dimensions. In the present paper we extend this result to the entire class of toric l.c.i.-singularities. Our proof makes use of Nakajima’s classification theorem and of some special techniques from toric and discrete geometry. 1

It is proved that whenever P is a prime ideal in a commutative Noetherian ring such that the P-adic and the P-symbolic topologies are equivalent, then the two topologies are equivalent linearly. Several explicit examples are calculated, in particular for all prime ideals corresponding to non-torsion points on nonsingular elliptic cubic curves.