26 research outputs found
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
Computational Approaches to Problems in Noncommutative Algebra -- Theory, Applications and Implementations
Noncommutative rings appear in several areas of mathematics. Most
prominently, they can be used to model operator equations, such as
differential or difference equations.
In the Ph.D. studies leading to this thesis, the focus was mainly on
two areas: Factorization in certain noncommutative domains and matrix
normal forms over noncommutative principal ideal domains.
Regarding the area of factorization, we initialize in this thesis a classification of noncommutative domains with
respect to the factorization properties of their elements. Such a
classification is well established in the area of commutative integral
domains. Specifically, we define conditions to identify so-called
finite factorization domains, and discover that the ubiquitous
G-algebras are finite factorization domains. We furthermore
realize a practical factorization algorithm
applicable to G-algebras, with minor assumptions on the underlying field. Since the generality of our algorithm
comes with the price of performance, we also study how it can be optimized for specific domains. Moreover, all of these factorization
algorithms are implemented.
However, it turns out that factorization
is difficult for many types of noncommutative rings. This observation leads to the adjunct
examination of noncommutative rings in the context of cryptography. In
particular, we develop a Diffie-Hellman-like key exchange protocol
based on certain noncommutative rings.
Regarding the matrix normal forms, we present a polynomial-time
algorithm of Las Vegas type to compute the Jacobson normal form of matrices over
specific domains. We will study the flexibility, as well as the
limitations of our proposal.
Another core contribution of this thesis consists of various implementations
to assist future researchers working with noncommutative
algebras. Detailed reports on all these programs and software-libraries are
provided. We furthermore develop a benchmarking tool called SDEval, tailored to the
needs of the computer algebra community. A description of this
tool is also included in this thesis