### 1. Introduction

This package, named GBNP for Gröbner Bases for Non-commutative Polynomials, is intended for computing in (associative) Non-commutative algebras with a finite presentation. Starting from a free algebra A on a finite number of generating variables, the reader can specify a finite set \$G\$ of polynomials in order to study the quotient algebra of \$A\$ by the (2-sided) ideal of \$A\$ generated by \$G\$.

#### 1.1 Installation

To install GBNP, first download it from http://www.win.tue.nl/~amc/pub/grobner/GBNP-0.9.4.tar.gz, then unpack `GBNP-0.9.4.tar.gz` in the `pkg` subdirectory of your GAP installation (or in the `pkg` subdirectory of any other GAP root directory, for example one added with the `-l` argument) with the following command: `tar -xvzf GBNP-0.9.4.tar.gz`.

GBNP is then loaded with the GAP command

 ``` gap> LoadPackage( "GBNP" ); ```

Those who want to download this documentation can find it at http://www.win.tue.nl/~amc/pub/grobner/GBNPdoc-0.9.4.tar.gz and extract it with `tar -xvzf GBNPdoc-0.9.4.tar.gz`. It is also included in the package.

#### 1.2 Introduction to the package

If you wish to compute a Gröbner basis, create a list of NPs (noncommutative polynomials in our format), see 2.1. This can be done either directly or by use of the transition functions described in Section 3.1. But beware: the algorithm for computing Gröbner bases is not guaranteed to terminate.

To run the standard algorithm use the functions from Section 3.4. With these functions, you can try and find a Gröbner basis. But also you can find a basis of monomials for the quotient algebra Q if the GB is found and the dimension of Q is finite with the functions in Section 3.5. For more advanced analysis of Q, such as a proof of finite or infinite dimensionality, or for determining its growth or its partial Hilbert series, use the functions from Section 3.6 .

There are two variants of the Gröbner basis algorithm, the truncated version in the homogeneous case, which is described in Section 2.6, which adds the functions described in Section 3.8 , and the version that gives an expression of the polynomials in terms of the original generators, described in 2.5, which adds the functions described in 3.7 .

Read the example files in Chapter A. for inspiration. The source of the files can be perused for auxiliary functions, which are often used in the main functions but not needed by a first time user.