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 =
"example05.g"
gap> ### authors Cohen & Gijsbers
gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
gap>
gap> ### Last change: August 22 2001.
gap> ### amc
<#GAPDoc Label=
"Example05">
<Section Label=
"Example05"><Heading>The gcd of some univariate polynomials</Heading>
A list of univariate polynomials is generated. The result of the
Gröbner basis computation on this list should be a single
monic polynomial, their gcd.
<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 single variable be printed as x by means of
<Ref Func=
"GBNP.ConfigPrint" Style=
"Text"/>
<Listing><![
CDATA[
gap> GBNP.ConfigPrint(
"x");
]]></Listing>
Now input the relations in NP format (see <Ref Sect=
"NP"/>). They will be
assigned to <C>KI</C>.
<Listing><![
CDATA[
gap> p0 := [[[1,1,1],[1,1],[1],[]],[1,2,2,1]];;
gap> p1 := [[[1,1,1,1],[1,1],[]],[1,1,1]];;
gap> KI := [p0,p1];;
gap> for i in [2..12] do
> h := AddNP(AddNP(KI[i],KI[i-1],1,3),
> AddNP(BimulNP([1],KI[i],[]),KI[i-1],2,1),3,-5);
> Add(KI,h);
> od;
]]></Listing>
The relations can be shown with <Ref Func=
"PrintNPList" Style=
"Text"/>:
<Listing><![
CDATA[
gap> PrintNPList(KI);
x^3 + 2x^2 + 2x + 1
x^4 + x^2 + 1
- 10x^5 + 3x^4 - 6x^3 + 11x^2 - 2x + 7
100x^6 - 60x^5 + 73x^4 - 128x^3 + 57x^2 - 76x + 25
- 1000x^7 + 900x^6 - 950x^5 + 1511x^4 - 978x^3 + 975x^2 - 486x + 103
10000x^8 - 12000x^7 + 12600x^6 - 18200x^5 + 14605x^4 - 13196x^3 + 8013x^2 - 2\
792x + 409
- 100000x^9 + 150000x^8 - 166000x^7 + 223400x^6 - 204450x^5 + 181819x^4 - 123\
630x^3 + 55859x^2 - 14410x + 1639
1000000x^10 - 1800000x^9 + 2150000x^8 - 2780000x^7 + 2765100x^6 - 2504340x^5 \
+ 1840177x^4 - 982264x^3 + 343729x^2 - 70788x + 6553
- 10000000x^11 + 21000000x^10 - 27300000x^9 + 34850000x^8 - 36655000x^7 + 342\
32300x^6 - 26732590x^5 + 16070447x^4 - 6878602x^3 + 1962503x^2 - 335534x + 262\
15
100000000x^12 - 240000000x^11 + 340000000x^10 - 437600000x^9 + 479700000x^8 -\
463408000x^7 + 381083200x^6 - 250919600x^5 + 124358069x^4 - 44189892x^3 + 106\
17765x^2 - 1551904x + 104857
- 1000000000x^13 + 2700000000x^12 - 4160000000x^11 + 5480000000x^10 - 6219000\
000x^9 + 6212580000x^8 - 5347676000x^7 + 3789374800x^6 - 2103269850x^5 + 87925\
4915x^4 - 266261734x^3 + 55222347x^2 - 7046418x + 419431
10000000000x^14 - 30000000000x^13 + 50100000000x^12 - 68240000000x^11 + 79990\
000000x^10 - 82533200000x^9 + 74033300000x^8 - 55790408000x^7 + 33925155700x^6\
- 16106037100x^5 + 5797814361x^4 - 1527768240x^3 + 278602281x^2 - 31541180x +\
1677721
- 100000000000x^15 + 330000000000x^14 - 595000000000x^13 + 843500000000x^12 -\
1021260000000x^11 + 1087222000000x^10 - 1012808600000x^9 + 804854300000x^8 - \
528013485000x^7 + 277993337300x^6 - 114709334310x^5 + 36188145143x^4 - 8434374\
466x^3 + 1372108031x^2 - 139586422x + 6710887
gap> Length(KI);
13
]]></Listing>
The Gröbner basis can now be calculated with
<Ref Func=
"SGrobner" Style=
"Text"/>:
<Listing><![
CDATA[
gap> GB := SGrobner(KI);;
#I number of entered polynomials is 13
#I number of polynomials after reduction is 1
#I End of phase I
#I End of phase II
#I List of todo lengths is [ 0 ]
#I End of phase III
#I G: Cleaning finished, 0 polynomials reduced
#I End of phase IV
#I The computation took 0 msecs.
]]></Listing>
Printed it looks like:
<Listing><![
CDATA[
gap> PrintNPList(GB);
x^2 + x + 1
]]></Listing>
</Section>
<#/GAPDoc>
