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 $q$-Weyl algebra, which are both viewed as a $\mathbb{Z}$-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 non–degeneracy conditions) for certain noncommutative algebras to have Poincaré–Birkhoff– Witt basis. We study theoretical properties of such G–algebras, con-cluding they are in some sense ”close to commutative”. We use the non–degeneracy conditions for practical study of certain deformations of Weyl algebras, quadratic and diffusion algebras. The famous Poincaré–Birkhoff–Witt (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 Bergman’s 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 ”Does this algebra have a PBW basis? ” 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 non–degeneracy conditions (Theorem 2.3), the second one led us to the G – and GR–algebras (3.4) and their properties (Theorem 4.7, 4.8), and the third one — to the notion of G–quantization and to the descrip-tion and classification of G–algebras 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