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Right Buchberger Algorithm over Bijective Skew PBW Extensions

Fajardo William

Fundamenta Informaticae

2021

Vol. 184, nr 2

83--105

In this paper we present a right version of the Buchberger algorithm over skew Poincaré-Birkhoff-Witt extensions (skew PBW extensions for short) defined by Gallego and Lezama [5]. This algorithm is an adaptation of the left case given in [3]. In particular, we developed a right version of the division algorithm and from this we built the right Grbner bases theory over bijective skew PBW extensions. The algorithms were implemented in the SPBWE library developed in Maple, this paper includes an application of these to the membership problem. The theory developed here is fundamental to complete the SPBWE library and thus be able to implement various homological applications that arise as result of obtaining the right Grbner bases over skew PBW extensions.

Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Fajardo William, Lezama Oswaldo

Fundamenta Informaticae

2019

Vol. 164, nr 1

41--59

In this short note we present an elementary matrix-constructive algorithmic proof of the Quillen-Suslin theorem for Ore extensions A := K[x; σ, δ], where K is a division ring, σ : K → K is a division ring automorphism and σ : K → K is a σ-derivation of K. It asserts that every finitely generated projective A-module is free. We construct a symbolic algorithm that computes the basis of a given finitely generated projective A-module. The algorithm is implemented in a computational package. Its efficiency is illustrated by four representative examples.

A Computational Maple Library for Skew PBW Extensions

Fajardo William

Fundamenta Informaticae

2019

Vol. 167, nr 3

159--191

In this paper we present a computational package developed for making computations involved in many homological applications of the Grbner theory of skew PBW extensions.