gap> ######################### BEGIN COPYRIGHT MESSAGE #########################
GBNP - computing Gröbner bases of noncommutative polynomials
Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem
Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group
at the Department of Mathematics and Computer Science of Eindhoven
University of Technology.
For acknowledgements see the manual. The manual can be found in several
formats in the doc subdirectory of the GBNP distribution. The
acknowledgements formatted as text can be found in the file chap0.txt.
GBNP is free software; you can redistribute it and/or modify it under
the terms of the Lesser GNU General
Public License as published by the
Free Software Foundation (FSF); either
version 2.1 of the License, or
(at your option) any later
version. For details, see the file
'LGPL' in
the doc subdirectory of the GBNP distribution or see the FSF
's own site:
https://www.gnu.org/licenses/lgpl.html
gap> ########################## END COPYRIGHT MESSAGE ##########################
gap> ### filename =
"example09.g"
gap> ### authors Cohen & Gijsbers
gap> ### vs 0.8.1
gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
gap>
gap> ### Last change: amc May 24 2007.
gap> ### dahg
<#GAPDoc Label=
"Example09">
<Section Label=
"Example09"><Heading>Tracing an example by Mora</Heading>
This example of a non-commutative Gröbner basis computation
is from page 18 of
<Q>An introduction to commutative and non-commutative Gröbner Bases</Q>, by
Teo Mora <Cite Key=
"TCS::Mora1994:131"/>.
The traced
version of the algorithm will be used.
The input is <M>\{xyx-y,yxy-y\}</M>.
The answer should be <M>\{yy-xy,yx-xy,xxy-y\}</M>.
<P/>
First load the package and set the standard infolevel <Ref
InfoClass=
"InfoGBNP" Style=
"Text"/> to 2 and the time infolevel <Ref
Func=
"InfoGBNPTime" Style=
"Text"/> to 1 (for more information about the info
level, see Chapter <Ref Chap=
"Info"/>).
<Listing><![
CDATA[
gap> LoadPackage(
"gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,2);
gap> SetInfoLevel(InfoGBNPTime,1);
]]></Listing>
Let the variables be printed as <M>x</M> and <M>y</M> instead of <M>a</M> and <M>b</M> by means of
<Ref Func=
"GBNP.ConfigPrint" Style=
"Text"/>
<Listing><![
CDATA[
gap> GBNP.ConfigPrint(
"x",
"y");
]]></Listing>
Next we input the relations in NP format (see Section <Ref Sect=
"NP"/>). They
will be assigned to <C>KI</C>.
<Listing><![
CDATA[
gap> xyx := [[[1,2,1],[2]],[1,-1]];;
gap> yxy := [[[2,1,2],[2]],[1,-1]];;
gap> KI:=[xyx,yxy];;
]]></Listing>
The relations can be shown with <Ref Func=
"PrintNPList" Style=
"Text"/>:
<Listing><![
CDATA[
gap> PrintNPList(KI);
xyx - y
yxy - y
]]></Listing>
The Gröbner basis with trace can now be calculated with
<Ref Func=
"SGrobnerTrace" Style=
"Text"/>:
<Listing><![
CDATA[
gap> GB := SGrobnerTrace(KI);
#I number of entered polynomials is 2
#I number of polynomials after reduction is 2
#I End of phase I
#I End of phase II
#I j =2
#I Current number of elements in todo is 1
#I j =3
#I Current number of elements in todo is 0
#I List of todo lengths is [ 2, 1, 0 ]
#I End of phase III
#I End of phase IV
#I The computation took 4 msecs.
[ rec( pol := [ [ [ 2, 1 ], [ 1, 2 ] ], [ 1, -1 ] ],
trace := [ [ [ ], 1, [ 2 ], -1 ], [ [ 2 ], 1, [ ], 1 ],
[ [ 1 ], 2, [ ], 1 ], [ [ ], 2, [ 1 ], -1 ] ] ),
rec( pol := [ [ [ 2, 2 ], [ 1, 2 ] ], [ 1, -1 ] ],
trace := [ [ [ 2 ], 1, [ ], -1 ], [ [ ], 1, [ 2 ], -1 ],
[ [ 2 ], 1, [ ], 1 ], [ [ ], 2, [ 1 ], 1 ], [ [ 1 ], 2, [ ], 1 ],
[ [ ], 2, [ 1 ], -1 ] ] ),
rec( pol := [ [ [ 1, 1, 2 ], [ 2 ] ], [ 1, -1 ] ],
trace := [ [ [ ], 1, [ ], 1 ], [ [ 1 ], 1, [ 2 ], 1 ],
[ [ 1, 2 ], 1, [ ], -1 ], [ [ 1, 1 ], 2, [ ], -1 ],
[ [ 1 ], 2, [ 1 ], 1 ] ] ) ]
]]></Listing>
The Gröbner basis can be printed with
<Ref Func=
"PrintNPListTrace" Style=
"Text"/>:
<Listing><![
CDATA[
gap> PrintNPListTrace(GB);
yx - xy
y^2 - xy
x^2y - y
]]></Listing>
The trace of the Gröbner basis can be printed with
<Ref Func=
"PrintTraceList" Style=
"Text"/>:
<Listing><![
CDATA[
gap> PrintTraceList(GB);
- G(1)y + yG(1) - G(2)x + xG(2)
- G(1)y + xG(2)
G(1) + xG(1)y - xyG(1) + xG(2)x - x^2G(2)
]]></Listing>
</Section>
<#/GAPDoc>
