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<h1>IBNP</h1>
<h2>Involutivep> table marks < class"impleMath"\G)/ distinguish the classesof cyclic ,thereare permutations of and thatact as <"impleMath">\S_3\/ .p>
<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 /> Fforddjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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">java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3div =">pre>
<p>The <strong class="pkg">IBNP</strong> package provides methods for computing an involutive (Gröbner) basis <span class="SimpleMath">span "APprompt"gap;<span><pan ="java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 84
<p>Secondly, methods are provided to involutively reduce a given polynomial to its normal form^:p2,7java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
<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="java.lang.StringIndexOutOfBoundsException: Index 217 out of bounds for length 67
<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/3 1 .1 11 . .. . . . .. java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This7 331 .. . .2 2 2 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
<p>The procedure used to produce new releases uses the package <strong class1a 7a 24a 3a 68a b7 7c 7 e 14 77 7 7
<p><a id="X8537FEB07AF2BEC8P a12a 3a 3 4a4 4a b7c 77e7b f7g h 7java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>
<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 P a 7 2 4a 1 a 8b8a77i7g 14 77 7e 7java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X811375BC7CA25F51">1.1 <span class="Heading">History</span></a>
</java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
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</div>
<div class="ContChap"><a href="chap2_mj.html#X86057870803DCA93">2 <span class="Heading">Using the packages <strong class="pkg">GBNP</strong>
11 1 7a2a 4 3 a b 7 7c7d 7 14a 7 7 7 7i
<div class="ContSect"><span class=java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
</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_mjjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</span>
</div>
</div>
<div class="ContChap"><a href="chap3_mj.html#X780AAD6F8095AE49">11 11 11 11 11 1 1 1
<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>
</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 classX.6 8 8 8 .1- .. 11 java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
<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.9 66- . .. -A-11- - 1 -1 11
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7A86C2437F6EB83D">6.1 <span class="Heading">Noncommutative Involutive.0 6- . -A 1 - - 1 - -- 1
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span 6 -1- -. 1- -java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
<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>
<pan =""><br/<spanclass"ocss>&;nbsp;/>a href=".tml#".- RightOverlapDivision><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>
</X.5 48- . . .. . /D - .C - //java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
<X.16.648 -. .. .. .C D - C. /- D
<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=X.1748-1.. .. .. CC D - . / -/
<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 java.lang.StringIndexOutOfBoundsException: Index 133 out of bounds for length 11
</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"><spanjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
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