Computer Algebra System SINGULAR:PLURAL and non-commutative Groebner bases in theory and applications

Sprache des Vortragstitels:

Englisch

Original Kurzfassung:

Being less than 40 years old, Gr\"obner bases win more and more attention of both mathematicians and (what is even more important), of other scientists. Also in the non--commutative algebra Gr\"obner bases play an important role and have their applications both to branches of mathematics and to various other sciences. Following B.~Sturmfels, who called the most fundamental applications of Gr\"obner bases in the commutative case \textit{Gr\"obner basics}, we introduce non--commutative Gr\"obner basics over $G$--algebras (a.k.a. PBW algebras). We present the properties of $G$--algebras and their impact on applications. A family of "Gr\"obner bases--based" algorithms is still growing, and among them are Ideal (resp. module) membership problem, Intersection with subrings (elimination of variables), Kernel of a module homomorphism, Kernel of a ring homomorphism, Algebraic relations between polynomials. Indeed, one can deal with many of such applications, using only one, flexible enough but also quite fast algorithm. We show the concept of a \textit{Gr\"obner engine}, comment its use in \textsc{Singular:Plural} and see the latest developments in the area, especially the very recent \textit{Slim Gr\"obner basis} algorithm by Brickenstein. We show, how Gr\"obner basics are implemented in \textsc{Singular:Plural} and which difficulties one has with related algorithms in the non--commutative case. Some examples will be computed live. Moreover, we will present the recently developed library for computations with $D$--modules \texttt{dmod.lib} (which uses heavy Gr\"obner bases computations) and show some timings. We will present some open problems and questions.

Sprache der Kurzfassung:

Englisch

Vortragstyp:

Eingeladener Vortrag an anderen Institutionen

Vortragsdatum:

25.11.2005

Vortragsort:

Spanien

Details zum Vortragsort:

44th IEEE Conference on Decision and Control and the European Control Conference Sevilla