# Gröbner basis computation algorithms

## Abstract

In this chapter, we introduce most of the basic objects and tools that will be used throughout the rest of this book. We start with some background on the notion of admissible order ℕ^{ n } and show how this may be used to define left Poincaré-Birkhoff-Witt (PBW) rings. The elements of these rings exhibit a behaviour which is similar to that of ordinary, commutative polynomials, allowing them to be approached computationally. Left PBW rings are actually very nice algebras — they are left noetherian domains — and their class includes a vast amount of important instances, such as enveloping algebras of finite dimensional Lie algebras and iterated Ore extensions, which cover most “important” quantum groups. We will put particular emphasis on the division algorithm for these rings, as this will play a fundamental role when we introduce Gröbner bases for one-sided and two-sided ideals. We show how Buchberger’s approach generalizes to the noncommutative case. In particular, we will see that Gröbner bases always exist and that reduced Gröbner bases are unique.

## Keywords

Partial Order Left Ideal Division Ring Division Algorithm Newton Diagram## Preview

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