Viktor Levandovskyy

Computational D-module theory with singular, comparison with other systems and two new algorithms

We present the new implementation of core functions for the computational D-module theory. It is realized as a library dmod.lib in the computer algebra system Singular. We show both theoretical advances, such as the LOT and checkRoot algorithms as well as the comparison of our implementation with other packages for D-modules in computer algebra systems kan/sm1, Asir and Macaulay. The comparison indicates, that our implementation is among the fastest ones. With our package we are able to solve several challenges in D-module theory and we demonstrate the answers to these problems.

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.

Plural, a non-commutative extension of singular: past, present and future

We describe the non–commutative extension of the computer algebra system Singular, called Plural. In the system, we provide rich functionality for symbolic computation within a wide class of non–commutative algebras. We discuss the computational objects of Plural, the implementation of main algorithms, various aspects of software engineering and numerous applications.

Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

In this paper we present an algorithm for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Grobner bases. We propose a general framework of Ore localizations of non-commutative G-algebras and show its merits and constructiveness. It allows us to handle, among others, common operator algebras with rational coefficients. We introduce the splitting of the computation of a normal form (like the Jacobson form over simple domain) for matrices over Ore localizations into the diagonalization (the computation of a diagonal form of a matrix) and the normalization (the computation of the normal form of a diagonal matrix). These ideas are also used for the computation of the Smith normal form in the commutative case. We give a special algorithm for the normalization of a diagonal matrix over the rational Weyl algebra and present counterexamples to its idea over rational shift and q-Weyl algebras. Our implementation of the algorithm in Singular:Plural relies on the fraction-free polynomial strategy, details of which will be described in the forthcoming article. It shows quite an impressive performance, compared with methods which directly use fractions. In particular, we experience quite a moderate swell of coefficients and obtain uncomplicated transformation matrices. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a large class of non-commutative algebras.

Intersection of ideals with non-commutative subalgebras

Computation of an intersection of a left ideal with a subalgebra, which is not fully investigated until now, is important for different areas of mathematics.We present an algorithm for the computation of the preimage of a left ideal under a morphism of non-commutative GR-algebras, and show both its abilities and limitations.The main computational tools are the elimination of variables by means of Gröbner bases together with the constructive treatment of opposite algebras and the utilization of a special bimodule structure.

Constructive D-Module Theory with Singular

We overview numerous algorithms in computational D-module theory together with the theoretical background as well as the implementation in the computer algebra system Singular. We discuss new approaches to the computation of Bernstein operators, of logarithmic annihilator of a polynomial, of annihilators of rational functions as well as complex powers of polynomials. We analyze algorithms for local Bernstein–Sato polynomials and also algorithms, recovering any kind of Bernstein–Sato polynomial from partial knowledge of its roots. We address a novel way to compute the Bernstein–Sato polynomial for an affine variety algorithmically. All the carefully selected nontrivial examples, which we present, have been computed with our implementation. We also address such applications as the computation of a zeta-function for certain integrals and revealing the algebraic dependence between pairwise commuting elements.

Principal intersection and bernstein-sato polynomial of an affine variety

We present a general algorithm for computing an intersection of a left ideal of an associative algebra over a field with a subalgebra, generated by a single element. We show applications of this algorithm in different algebraic situations and describe our implementation in Singular. Among other, we use this algorithm in computational D-module theory for computing e.g. the Bernstein-Sato polynomial of a single polynomial with several approaches. We also present a new method, having no analogues yet, for the computation of the Bernstein-Sato polynomial of an affine variety. Also, we provide a new proof of the algorithm by Briançon-Maisonobe for the computation of the s-parametric annihilator of a polynomial.

On noncommutative finite factorization domains

A domain

A Symbolic Approach to Generation and Analysis of Finite Difference Schemes of Partial Differential Equations

R

Factorization of \( \mathbb {Z}\)-Homogeneous Polynomials in the First q-Weyl Algebra

is said to have the finite factorization property if every nonzero non-unit element of