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<?xml version="1.0" encoding="UTF-8"?>

<Chapter Label="chap-intro">

<Heading>Introduction</Heading>

The &IBNP; package provides methods for computing an involutive (&Grob;)
basis <M>B</M> for an ideal <M>J</M> over a polynomial ring
<M>\mathbb{R}</M> in both the commutative and noncommutative cases.
Secondly, methods are provided to involutively reduce a given polynomial
to its normal form in <M>\mathbb{R}/J</M>.

This package was started using &GAP; 4.13.1
and first released with &GAP; 4.15.0.


The package is loaded with the command
<Example>
<![CDATA[
gap> LoadPackage( "ibnp" );
]]>
</Example>


The package may be obtained as a compressed <Code>.tar</Code> file
<File>IBNP-0.16.tar.gz</File> from the GitHub release site:
<URL>https://github.com/gap-packages/ibnp/releases/tag/v0.16</URL>.
<Index>GitHub repository</Index>
The package also has a GitHub repository at:
<URL>https://github.com/gap-packages/ibnp</URL>.


Once the package is loaded, the manual <Code>doc/manual.pdf</Code>
can be found in the documentation folder.
The <Code>html</Code> versions, with or without MathJax,
may be rebuilt as follows:

<Example>
<![CDATA[
gap> ReadPackage( "ibnp", "makedoc.g" );
]]>
</Example>


It is possible to check that the package has been installed correctly
by running the test files (this terminates the &GAP; session):

<Example>
<![CDATA[
gap> TestPackage( "ibnp" );
Architecture: . . . . .
testing: . . . . .
. . .
#I No errors detected while testing
]]>
</Example>


<Section Label="sec-history">
<Heading>History</Heading>

The theoretical basis behind this package is the Ph.D. thesis
"Noncommutative Involutive Bases" of Gareth Evans in 2005
<Cite Key='gareth-thesis'/>.
(The main concepts and results may also be found in the papers
<Cite Key='Beaumont'/> and <Cite Key='EW-JSC'/>.)
We quote here the summary from that thesis.


<E>The theory of &Grob; Bases originated in the work of Buchberger
<Cite Key='buchberger'/> and is now considered to be one of the
most important and useful areas of symbolic computation.
A great deal of effort has been put into improving Buchberger's algorithm
for computing a &Grob; Basis,
and indeed in finding alternative methods of computing &Grob; Bases.
Two of these methods include the &Grob; Walk method <Cite Key='onthewalk'/>
and the computation of Involutive Bases <Cite Key='yuyu'/>.</E>

<E>By the mid 1980's, Buchberger's work had been generalised for
noncommutative polynomial rings by Bergman <Cite Key='bergman'/>
and Mora <Cite Key='mora'/>.
This thesis provides the corresponding generalisation
for Involutive Bases and (to a lesser extent) the &Grob; Walk,
with the main results being as follows.
</E>

<List>
<Item>
<E>Algorithms for several new noncommutative involutive divisions
are given, including strong; weak; global and local divisions.</E>
</Item>
<Item>
<E>An algorithm for computing a noncommutative Involutive Basis is given.
When used with one of the aforementioned involutive divisions,
it is shown that this algorithm returns a noncommutative &Grob; Basis
on termination.</E>
</Item>
<Item>
<E>An algorithm for a noncommutative &Grob; Walk is given,
in the case of conversion between two harmonious monomial orderings.
It is shown that this algorithm generalises to give an algorithm
for performing a noncommutative Involutive Walk,
again in the case of conversion between two harmonious monomial orderings.</E>
</Item>
<Item>
<E>Two new properties of commutative involutive divisions are introduced
(stability and extendibility), respectively ensuring the termination
of the Involutive Basis algorithm and the applicability
(under certain conditions) of homogeneous methods of computing
Involutive Bases.</E>
</Item>
</List>

<E>Source code for an initial implementation of an algorithm to compute
noncommutative Involutive Bases is provided in Appendix B.
This source code, written using ANSI C and a series of libraries
(AlgLib) provided by MSSRC
forms part of a larger collection of programs providing examples for this thesis,
including implementations of the commutative and noncommutative
&Grob; Basis algorithms <Cite Key='buchberger'/>, <Cite Key='mora'/>;
the commutative Involutive Basis algorithm for the Pommaret and Janet
involutive divisions <Cite Key='yuyu'/>;
and the Knuth-Bendix critical pairs completion algorithm for monoid rewrite systems <Cite Key='knuth-bendix'/>.</E>

The implementations described in the last paragraph formed a package
<E>Involutive</E> in 2005.
This was based on libraries developed by the
<E>Multidisciplinary Software Systems Research Corporation</E> (MSSRC)
which apparently no longer exists.
This software was provided for the research by Larry Lambe
who was an Honorary Professor in the Mathematics Department
at Bangor at that time.
(For an example of his work, see <Cite Key='lambe-radford'/>.)

It has long been our intention to make these algorithms available
to the wider symbolic computation community,
and this &GAP; package is the result.
Involutive Bases are constructed, but &Grob; Walks will have to wait
until a later version.
In place of the AlgLib library functions we use the noncommutative
polynomial operations provided by the <Package>GBNP</Package>
<Cite Key='GBNP'/> package.

</Section>

</Chapter>

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