Non-commutative Elimination in Ore Algebras Proves Multivariate Identities

International audienceMany computations involving special functions, combinatorial sequences or their $q$-analogues can be performed using linear operators and simple arguments on the dimension of related vector spaces. In this article, we develop a theory of~$\partial$-finite sequences and functions which provides a unified framework to express algorithms for computing sums and integrals and for the proof or discovery of multivariate identities. This approach is vindicated by an implementation

Elimination of Variables in Linear Solvable Polynomial Algebras and ∂-Holonomicity

AbstractLet k be a field of characteristic 0. Based on the Gelfand–Kirillov dimension computation of modules over solvable polynomial k-algebras, where solvable polynomial algebras are in the sense of A. Kandri-Rody and V. Weispfenning (1990, J. Symbolic Comput.9, 1–26), we prove that the elimination lemma, obtained from D. Zeilberger (1990, J. Comput. Appl. Math.32, 321–368) by using holonomic modules over the Weyl algebra An(k) and used in the automatic proving of special function identities, holds for a class of solvable polynomial k-algebras without any “holonomicity” restriction. This opens a way to the solution of the extension/contraction problem stemming from the automatic proving of multivariate identities with respect to the ∂-finiteness in the sense of F. Chyzak and B. Salvy (1998, J. Symbolic Comput.26, 187–227). It also yields a ∂-holonomicity so that automatic proving of multivariate identities may be dealt with by manipulating polynomial function coefficients instead of rational functions

Exact linear modeling using Ore algebras

Linear exact modeling is a problem coming from system identification: Given a set of observed trajectories, the goal is find a model (usually, a system of partial differential and/or difference equations) that explains the data as precisely as possible. The case of operators with constant coefficients is well studied and known in the systems theoretic literature, whereas the operators with varying coefficients were addressed only recently. This question can be tackled either using Gr\"obner bases for modules over Ore algebras or by following the ideas from differential algebra and computing in commutative rings. In this paper, we present algorithmic methods to compute "most powerful unfalsified models" (MPUM) and their counterparts with variable coefficients (VMPUM) for polynomial and polynomial-exponential signals. We also study the structural properties of the resulting models, discuss computer algebraic techniques behind algorithms and provide several examples

Basic Module Theory over Non-Commutative Rings with Computational Aspects of Operator Algebras

The present text surveys some relevant situations and results where basic Module Theory interacts with computational aspects of operator algebras. We tried to keep a balance between constructive and algebraic aspects.Comment: To appear in the Proceedings of the AADIOS 2012 conference, to be published in Lecture Notes in Computer Scienc

Computing diagonal form and Jacobson normal form of a matrix using Gr\"obner bases

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative $G$-algebras and show its merits. In particular, for a given matrix $M$ we provide an algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in SINGULAR:PLURAL and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras

Some homological properties of skew PBW extensions

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations include a wide variety of rings and algebras of interest for modern mathematical physics such as PBW extensions, well known classes of Ore algebras, operator algebras, diffusion algebras, quantum algebras, quadratic algebras in 3-variables, skew quantum polynomials, among many others. We estimate the global, Krull and Goldie dimensions, and also Quillen's K-groups

A Fast Approach to Creative Telescoping

In this note we reinvestigate the task of computing creative telescoping relations in differential-difference operator algebras. Our approach is based on an ansatz that explicitly includes the denominators of the delta parts. We contribute several ideas of how to make an implementation of this approach reasonably fast and provide such an implementation. A selection of examples shows that it can be superior to existing methods by a large factor.Comment: 9 pages, 1 table, final version as it appeared in the journa