64 research outputs found
Obstructions to Genericity in Study of Parametric Problems in Control Theory
We investigate systems of equations, involving parameters from the point of
view of both control theory and computer algebra. The equations might involve
linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference
as well as more complicated ones, which act trivially on the parameters. Such a
system can be identified algebraically with a certain left module over a
non-commutative algebra, where the operators commute with the parameters. We
develop, implement and use in practice the algorithm for revealing all the
expressions in parameters, for which e.g. homological properties of a system
differ from the generic properties. We use Groebner bases and Groebner basics
in rings of solvable type as main tools. In particular, we demonstrate an
optimized algorithm for computing the left inverse of a matrix over a ring of
solvable type. We illustrate the article with interesting examples. In
particular, we provide a complete solution to the "two pendula, mounted on a
cart" problem from the classical book of Polderman and Willems, including the
case, where the friction at the joints is essential . To the best of our
knowledge, the latter example has not been solved before in a complete way.Comment: 20 page
Factorization of Z-homogeneous polynomials in the First (q)-Weyl Algebra
We present algorithms to factorize weighted homogeneous elements in the first
polynomial Weyl algebra and -Weyl algebra, which are both viewed as a
-graded rings. We show, that factorization of homogeneous
polynomials can be almost completely reduced to commutative univariate
factorization over the same base field with some additional uncomplicated
combinatorial steps. This allows to deduce the complexity of our algorithms in
detail. Furthermore, we will show for homogeneous polynomials that
irreducibility in the polynomial first Weyl algebra also implies irreducibility
in the rational one, which is of interest for practical reasons. We report on
our implementation in the computer algebra system \textsc{Singular}. It
outperforms for homogeneous polynomials currently available implementations
dealing with factorization in the first Weyl algebra both in speed and elegancy
of the results.Comment: 26 pages, Singular implementation, 2 algorithms, 1 figure, 2 table
PBW bases, non-degeneracy conditions and applications
Abstract. We establish an explicit criteria (the vanishing of nonâdegeneracy conditions) for certain noncommutative algebras to have PoincareÌâBirkhoffâ Witt basis. We study theoretical properties of such Gâalgebras, con-cluding they are in some sense âclose to commutativeâ. We use the nonâdegeneracy conditions for practical study of certain deformations of Weyl algebras, quadratic and diffusion algebras. The famous PoincareÌâBirkhoffâWitt (or, shortly, PBW) theorem, which ap-peared at first for universal enveloping algebras of finite dimensional Lie algebras ([7]), plays an important role in the representation theory as well as in the the-ory of rings and algebras. Analogous theorem for quantum groups was proved by G. Lusztig and constructively by C. M. Ringel ([6]). Many authors have proved the PBW theorem for special classes of noncom-mutative algebras they are dealing with ([17], [18]). Usually one uses Bergmanâs Diamond Lemma ([4]), although it needs some preparations to be done before ap-plying it. We have defined a class of algebras where the question âDoes this algebra have a PBW basis? â reduces to a direct computation involving only basic polyno-mial arithmetic. In this article, our approach is constructive and consists of three tasks. Firstly, we want to find the necessary and sufficient conditions for a wide class of algebras to have a PBW basis, secondly, to investigate this class for useful properties, and thirdly, to apply the results to the study of certain special types of algebras. The first part resulted in the nonâdegeneracy conditions (Theorem 2.3), the second one led us to the G â and GRâalgebras (3.4) and their properties (Theorem 4.7, 4.8), and the third one â to the notion of Gâquantization and to the descrip-tion and classification of Gâalgebras among the quadratic and diffusion algebras
Quantum Drinfeld Hecke Algebras
We consider finite groups acting on quantum (or skew) polynomial rings.
Deformations of the semidirect product of the quantum polynomial ring with the
acting group extend symplectic reflection algebras and graded Hecke algebras to
the quantum setting over a field of arbitrary characteristic. We give necessary
and sufficient conditions for such algebras to satisfy a Poincare-Birkhoff-Witt
property using the theory of noncommutative Groebner bases. We include
applications to the case of abelian groups and the case of groups acting on
coordinate rings of quantum planes. In addition, we classify graded
automorphisms of the coordinate ring of quantum 3-space. In characteristic
zero, Hochschild cohomology gives an elegant description of the
Poincare-Birkhoff-Witt conditions.Comment: 29 pages. Last example corrected; some indices in the last theorem
were accidentally transposed and now appear in correct orde
Certifying solutions to square systems of polynomial-exponential equations
Smale's alpha-theory certifies that Newton iterations will converge
quadratically to a solution of a square system of analytic functions based on
the Newton residual and all higher order derivatives at the given point. Shub
and Smale presented a bound for the higher order derivatives of a system of
polynomial equations based in part on the degrees of the equations. For a given
system of polynomial-exponential equations, we consider a related system of
polynomial-exponential equations and provide a bound on the higher order
derivatives of this related system. This bound yields a complete algorithm for
certifying solutions to polynomial-exponential systems, which is implemented in
alphaCertified. Examples are presented to demonstrate this certification
algorithm.Comment: 20 page
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