[MathJax on]

# GBNP

## computing Gröbner bases of noncommutative polynomials

1.0.5

9 March 2022

**
A.M. Cohen
**

Email: A.M.Cohen@tue.nl

**
J.W. Knopper
**

Email: J.W.Knopper@tue.nl

**Address:**

TU/e,

POB 513, 5600 MB Eindhoven, the Netherlands

### Abstract

We provide algorithms, written in the **GAP** 4 programming language, for computing Gröbner bases of non-commutative polynomials, and some variations, such as a weighted and truncated version and a tracing facility. In addition, there are algorithms for analyzing the quotient of a non-commutative polynomial algebra by a 2-sided ideal generated by a set of polynomials whose Gröbner basis has been determined and for computing quotient modules of free modules over quotient algebras.

The notion of algorithm is interpreted loosely: in general one cannot expect a non-commutative Gröbner basis algorithm to terminate, as it would imply solvability of the word problem for finitely presented (semi)groups.

This documentation gives a short description of the mathematical content, explains the functions of the package, and provides more than twenty worked out examples.

### Copyright

© 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group at the Department of Mathematics and Computer Science of Eindhoven University of Technology.

### Acknowledgements

The package is based on an earlier version by Rosane Ushirobira.

The bulk of the package is written by Arjeh M. Cohen and Dié A.H. Gijsbers.

The theory is mainly taken from literature by Teo Mora [Mor94] and Edward L. Green [Gre99].

From Version 0.8.3 on the package has three additional files (`fincheck.g`

, `tree.g`

`graphs.g`

) with routines for finding the Hilbert function and testing finite dimensionality when given a Gröbner basis by Chris Krook [Kro03], based on work by Victor Ufnarovski [Ufn89].

From Version 0.9 on the package is enriched with support for fields implemented in GAP and additional prefix rules for quotient modules, as well as some speed improvements by Jan Willem Knopper. Knopper has also formatted the documentation in GAPDoc [LN06].

From Version 1.0 on the package is extended with NMO (for Noncommutative Monomial Orderings) by Randall Cone. This enables the GBNP user to choose a wider selection of monomial orderings than the standard one built into GBNP itself. Documentation on NMO can be found in the NMO manual [Con10].