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<h1>IBNP</h1>


<h2>Involutive Bases for Noncommutative Polynomials</h2>

<p>
    0.17</p>

<p>
    11 September 2025
  </p>

</div>
<p><b>
    Gareth A. Evans



  </b>
<br />Email: <span class="URL"><a href="mailto:gareth@mathemateg.com">gareth@mathemateg.com</a></span>
<br />Address: <br />Ysgol y Creuddyn <br /> Ffordd Derwen, Bae Penrhyn <br /> Llandudno, LL30 3LB <br /> U.K.<br />
</p><p><b>
    Christopher D. Wensley



  </b>
<br />Email: <span class="URL"><a href="mailto:cdwensley.maths@btinternet.com">cdwensley.maths@btinternet.com</a></span>
<br />Homepage: <span class="URL"><a href="https://github.com/cdwensley">https://github.com/cdwensley</a></span>
</p>

<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The <strong class="pkg">IBNP</strong> package provides methods for computing an involutive (Gröbner) basis <span class="SimpleMath">\(B\)</span> for an ideal <span class="SimpleMath">\(J\)</span> over a polynomial ring <span class="SimpleMath">\(R\)</span> in both the commutative and noncommutative cases.</p>

<p>Secondly, methods are provided to involutively reduce a given polynomial to its normal form in <span class="SimpleMath">\(R/J\)</span>.</p>

<p>Bug reports, comments, suggestions for additional features, and offers to implement some of these, will all be very welcome.</p>

<p>Please submit any issues at <span class="URL"><a href="https://github.com/gap-packages/ibnp/issues/">https://github.com/gap-packages/ibnp/issues/</a></span> or send an email to the second author at <span class="URL"><a href="mailto:cdwensley.maths@btinternet.com">cdwensley.maths@btinternet.com</a></span>.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2024-2025, Gareth Evans and Chris Wensley.</p>

<p>The <strong class="pkg">IBNP</strong> package is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This documentation was prepared with the <strong class="pkg">GAPDoc</strong> <a href="chapBib_mj.html#biBGAPDoc">[LN17]</a> and <strong class="pkg">AutoDoc</strong> <a href="chapBib_mj.html#biBAutoDoc">[GH23]</a> packages.</p>

<p>The procedure used to produce new releases uses the package <strong class="pkg">GitHubPagesForGAP</strong> <a href="chapBib_mj.html#biBGitHubPagesForGAP">[Hor19]</a> and the package <strong class="pkg">ReleaseTools</strong>.</p>

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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X811375BC7CA25F51">1.1 <span class="Heading">History</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X86057870803DCA93">2 <span class="Heading">Using the packages <strong class="pkg">GBNP</strong>
and <strong class="pkg">NMO</strong></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X783C6EC87988B533">2.1 <span class="Heading">Noncommutative polynomials (NPs)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7E4277497D877661">2.2 <span class="Heading">Gröbner Bases</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7F82A3608248CD31">2.3 <span class="Heading">Orderings for monomials</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3_mj.html#X780AAD6F8095AE49">3 <span class="Heading">Commutative Involutive Bases</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7BBDABD3799443BB">3.1 <span class="Heading">Reduction Paths</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B5623E3821CC0D0">3.1-1 <span class="Heading">An Example</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7E43F2087BC8B4F9">3.2 <span class="Heading">Commutative Involutive Divisions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85861B017AEEC50B">3.2-1 <span class="Heading">Example</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83A3B3F77C712DA1">3.2-2 <span class="Heading">Selecting a Division</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X785D706A86DD7343">3.2-3 <span class="Heading">Selecting an Ordering</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X82712BA57EBE9170">3.2-4 PommaretDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8756720A86A6B125">3.2-5 ThomasDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E214DDF794BB14D">3.2-6 JanetDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8781FDB7865FA48B">3.2-7 DivisionRecord</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79F5892C80AE2667">3.2-8 IPolyReduce</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A36AE827C4012FF">3.2-9 LoggedIPolyReduce</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C58A339832877E9">3.2-10 IAutoreduce</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X864907F987701716">3.3 <span class="Heading">Computing a Commutative Involutive Basis</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7ACAA0847CC0DBCC">3.3-1 <span class="Heading">Prolongations and Autoreduction</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B60A306820D4ED2">3.3-2 InvolutiveBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X831D45437DE37177">3.3-3 <span class="Heading">A more detailed example</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X791740DF84B742A2">3.3-4 <span class="Heading">Using homogeneous polynomials</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X872783907DFA29B7">4 <span class="Heading">Functions for Noncommutative Monomials</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X846C3B0F79265278">4.1 <span class="Heading">Basic functions for monomials</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X84F106BB8093FCAE">4.1-1 <span class="Heading">Predefined algebras</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7D53D8657AEDFEB2">4.1-2 PrintNM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7AD4CF167E6B7D2E">4.1-3 NM2GM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8719E2857E26325C">4.1-4 GM2NM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F72641C8441204E">4.1-5 PrefixNM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8046DF397ACA0E5E">4.1-6 SuffixPrefixPosNM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82916CB37D346978">4.1-7 SubwordPosNM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X83CF80DD7CD5F166">4.1-8 LeadVarNM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7CECFE0C86895946">4.1-9 DivNM</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap5_mj.html#X7BD27C5585EF8629">5 <span class="Heading">Functions for Noncommutative Polynomials</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X80FC94957D03EEA6">5.1 <span class="Heading">Basic functions for polynomials</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A1E54F279CCCF65">5.1-1 MaxDegreeNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7903A443865A3471">5.1-2 ScalarMulNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7996395279064998">5.1-3 LtNPoly</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X79B2E02082C8799E">5.1-4 LowestLeadMonomialPosNP</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X797E483A84214975">6 <span class="Heading">Noncommutative Involutive Bases</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7A86C2437F6EB83D">6.1 <span class="Heading">Noncommutative Involutive Divisions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8593BCDB8402C46C">6.1-1 LeftDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X784AF6B87B2B5E5D">6.1-2 RightDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A979BF38311024C">6.1-3 LeftOverlapDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X83CE05CF7CB18611">6.1-4 RightOverlapDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X83A3B3F77C712DA1">6.1-5 <span class="Heading">Selecting a Division</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X86FAAD527E20A573">6.1-6 DivisionRecordNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X828DA2AE844847E9">6.1-7 IPolyReduceNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X78935EBD85A02F3F">6.1-8 LoggedIPolyReduceNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7EBFCE307BE928BA">6.1-9 VerifyLoggedRecordNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X8189DEDD87CE1667">6.1-10 IAutoreduceNP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X80C3BE018688AFB7">6.2 <span class="Heading">Computing a Noncommutative Involutive Basis</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7A71E4CD7B43726B">6.2-1 InvolutiveBasisNP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7DB2608C86CA3B04">6.3 <span class="Heading">The Disjoint Cones Conditions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7D87860878548EF2">6.3-1 StrongLeftOverlapDivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X819FE8F87ACDB19C">6.3-2 StrongRightOverlapDivision</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
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