Non-commutative Groebner basics: from the theory to the implementation in the Computer Algebra System SINGULAR:PLURAL
Sprache des Vortragstitels:
Englisch
Original Kurzfassung:
Following Sturmfels, who once proposed to treat the most fundamental applications of Gr\"obner bases in the commutative case as a collection of \textit{Gr\"obner basics}, we introduce non--commutative Gr\"obner basics over $G$--algebras (a.k.a. PBW algebras). We go through the various (ring--theoretical, homological etc) properties of $G$--algebras and point out the properties, which can be proved by constructive means. Among the non--commutative Gr\"obner basics, there are Ideal (resp. module) membership problem, Intersection with subrings (elimination of variables), Intersection of ideals (resp. submodules), Quotient and Saturation of ideals, Kernel of a module homomorphism, Kernel of a ring homomorphism, Algebraic relations between polynomials. All these applications are implemented in \textsc{Singular:Plural}, we will comment some aspects of their use and compute some examples live. Moreover, there are several methods for computing syzygies and free resolutions of modules implemented. However, there are important applications, arising in the non--commutative case, such as Two--sided Gr\"obner basis for bimodules, Gel'fand--Kirillov dimension for modules, Operations with opposite and enveloping algebras, Preimage of left ideal under a morphism of $GR$--algebras, Annihilators of finite--dimensional modules. We show, how these algorithms are implemented and which difficulties one has with related algorithms. We will present some open theoretical and algorithmical problems as well as directions of future research.
Forschungs-