<p>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 <span class="SimpleMath">A</span> on a finite number of generating variables, the reader can specify a finite set <span class="SimpleMath">G</span> of polynomials in these variables, in order to study the quotient algebra of <span class="SimpleMath">A</span> by the (2-sided) ideal of <span class="SimpleMath">A</span> generated by <span class="SimpleMath">G</span>.</p>
<p>This documentation gives a short description of the mathematical content in Chapter <a href="chap2.html#X7BBCB13F82ACC213"><span class="RefLink">2</span></a>, explains the functions of the package in Chapter <a href="chap3.html#X86FA580F8055B274"><span class="RefLink">3</span></a>, and provides more than twenty four worked out examples in Appendix <a href="chapA.html#X7A489A5D79DA9E5C"><span class="RefLink">A</span></a>. It is available as an HTML document at <span class="URL"><a href="https://gap-packages.github.io/gbnp/doc/chap0.html">https://gap-packages.github.io/gbnp/doc/chap0.html</a></span>.</p>
<p>To install GBNP, first download <code class="file">GBNP-1.1.0.tar.gz</code> from <span class="URL"><a href="https://gap-packages.github.io/gbnp/">https://gap-packages.github.io/gbnp/</a></span>, then unpack <code class="file">GBNP-1.1.0.tar.gz</code> in the <code class="code">pkg</code> subdirectory of your <strong class="pkg">GAP</strong> installation (or in the <code class="code">pkg</code> subdirectory of any other <strong class="pkg">GAP</strong> root directory, for example one added with the <code class="code">-l</code> argument) with the following command: <code class="code">tar -xvzf GBNP-1.1.0.tar.gz</code>.</p>
<h4>1.2 <span class="Heading">Using the package</span></h4>
<p>If you wish to compute a Gröbner basis, create a list of NPs (non-commutative polynomials in NP format), as described in Section <a href="chap2.html#X7FDF3E5E7F33D3A2"><span class="RefLink">2.1</span></a>. This can be done either directly or by use of the transition functions described in Section <a href="chap3.html#X81ABB91B79E00229"><span class="RefLink">3.1</span></a>. To run the standard algorithm use the functions from Section <a href="chap3.html#X81381B2D83D2B9A9"><span class="RefLink">3.4</span></a>. With these functions, you can try and find a Gröbner basis. The word try is included because the algorithm for computing Gröbner bases is not guaranteed to terminate. Printing issues for polynomials in NP format are discussed in Section <a href="chap3.html#X78F44B01851B1020"><span class="RefLink">3.2</span></a>. If the Gröbner basis is found and the dimension of the quotient algebra <span class="SimpleMath">Q</span> (see Section <a href="chap2.html#X85A91A467FF1DE45"><span class="RefLink">2.9</span></a>) is finite, you can find a basis of monomials for <span class="SimpleMath">Q</span> with the functions in Section <a href="chap3.html#X7F387F7780425B9A"><span class="RefLink">3.5</span></a>. For a more advanced analysis of <span class="SimpleMath">Q</span>, such as a proof of finite or infinite dimensionality, or for determining its growth or its partial Hilbert series, use the functions from Section <a href="chap3.html#X79FE4A3983E2329F"><span class="RefLink">3.6</span></a> .</p>
<p>There are three variants of the Gröbner basis algorithm, the truncated version, the trace version, and the module version. In the (weighted) homogeneous case (described in Section <a href="chap2.html#X78CF5C44879D34B6"><span class="RefLink">2.6</span></a>), the truncated version, given by the functions described in Section <a href="chap3.html#X7E4E3AD07B2465F9"><span class="RefLink">3.8</span></a>, computes the part of a Gröbner basis up to an indicated weight. The trace version (described in Section <a href="chap2.html#X8739B6547BC89505"><span class="RefLink">2.5</span></a>), given by the functions described in Section <a href="chap3.html#X7BA5CAA07890F7AA"><span class="RefLink">3.7</span></a>, computes an expression of the polynomials of the Gröbner basis found in terms of the original generators. The module version (described in Sections <a href="chap2.html#X7B27E2D1784538DE"><span class="RefLink">2.2</span></a>, <a href="chap2.html#X86F1F4EE7D4D06B7"><span class="RefLink">2.7</span></a>, and <a href="chap2.html#X80DAE0A97CFC95DD"><span class="RefLink">2.8</span></a>), given by the functions described in Section <a href="chap3.html#X8706DD3287E82019"><span class="RefLink">3.9</span></a>, computes a Gröbner basis for a submodule of a free <span class="SimpleMath">Q</span>-module of finite rank.</p>
<p>Read the example files in Chapter <a href="chapA.html#X7A489A5D79DA9E5C"><span class="RefLink">A</span></a> for inspiration. The source of the files can be perused for auxiliary functions, which are often used in the main functions but not deemed necessary for a first time user.</p>
<p>The reports <a href="chapBib.html#biBCohenGijsbersEtAl2007">[Coh07]</a>, <a href="chapBib.html#biBKrook2003">[Kro03]</a>, and <a href="chapBib.html#biBKnopper2004">[Kno04]</a> can be downloaded from the web at these addresses:</p>
<p>The report <q>Non-commutative polynomial computations</q>, by Arjeh M. Cohen (with support of Dié Gijsbers, Jan Willem Knopper, and Chris Krook) can be downloaded from <span class="URL"><a href=" http://mathdox.org/products/gbnp/gbnp.pdf"> http://mathdox.org/products/gbnp/gbnp.pdf</a></span>.</p>
<p>The report <q>Dimensionality of quotient algebras</q>, by Chris Krook can be downloaded from <span class="URL"><a href=" http://mathdox.org/products/gbnp/dqa.pdf"> http://mathdox.org/products/gbnp/dqa.pdf</a></span>.</p>
<p>The report <q>GBNP and vector enumeration</q>, by Jan Willem Knopper can be downloaded from <span class="URL"><a href=" http://mathdox.org/products/gbnp/knopper.pdf"> http://mathdox.org/products/gbnp/knopper.pdf</a></span>.</p>
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