Non-Commutative Gebauer-Moeller Criteria

TL;DR

This paper extends the Gebauer-Moeller criteria to non-commutative polynomial rings, enabling more efficient Buchberger's Algorithm by identifying and avoiding unnecessary obstructions in free associative algebras.

Contribution

It introduces a novel adaptation of Gebauer-Moeller criteria for non-commutative rings, improving the efficiency of Buchberger's Algorithm in these settings.

Findings

Almost all unnecessary obstructions are detected during Buchberger's procedure.

The new criteria significantly reduce the number of critical pairs processed.

Experiments demonstrate improved efficiency in non-commutative polynomial ideal computations.

Abstract

For an efficient implementation of Buchberger's Algorithm, it is essential to avoid the treatment of as many unnecessary critical pairs or obstructions as possible. In the case of the commutative polynomial ring, this is achieved by the Gebauer-Moeller criteria. Here we present an adaptation of the Gebauer-Moeller criteria for non-commutative polynomial rings, i.e. for free associative algebras over fields. The essential idea is to detect unnecessary obstructions using other obstructions with or without overlap. Experiments show that the new criteria are able to detect almost all unnecessary obstructions during the execution of Buchberger's procedure.