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: http://www.gnu.org/licenses/lgpl.html gap> ########################## END COPYRIGHT MESSAGE ##########################

gap> ### filename = "example25.g" gap> ### author amc gap> ### amc last edited 1 June 2007

<#GAPDoc Label="Example25">
Baur and Draisma's transformations
gap> LoadPackage("GBNP","0",false);; gap> SetInfoLevel(InfoGBNP,0); gap> SetInfoLevel(InfoGBNPTime,0);

The paper by Baur and Draisma uses the computation of a quotient algebra of dimension 37, which we repeat here. The set of equations, after specialisation of the scalars to 1, is as follows.

gap> KI := [ [[[2,2]],[1]], > [[[1,1]],[1]], > [[[3,3]],[1]], > [[[1,2,1],[1]],[1,-1]], > [[[2,1,2],[2]],[1,-1]], > [[[3,2,3],[3]],[1,-1]], > [[[2,3,2],[2]],[1,-1]], > [[[1,3,1],[1]],[1,-1]], > [[[3,1,3],[3]],[1,-1]], > [[[1,2,3,1,2,3,1],[1,3,2,1,3,2,1],[1]],[1,1,-1]], > [[[3,1,2,3,1,2,3],[3,2,1,3,2,1,3],[3]],[1,1,-1]], > [[[2,3,1,2,3,1,2],[2,1,3,2,1,3,2],[2]],[1,1,-1]], > ];; gap> PrintNPList(KI); b^2 a^2 c^2 aba - a bab - b cbc - c bcb - b aca - a cac - c abcabca + acbacba - a cabcabc + cbacbac - c bcabcab + bacbacb - b

We carry out a traced Gröbner basis computation by use of , and form the usual Gröbner basis by extracting the polynomials from the traced polynomials using the field indicator .pol. gap> GBT := SGrobnerTrace(KI);; gap> GB := List([1..Length(GBT)], i -> GBT[i].pol);;

The dimension of the quotient algebra is computable with . gap> DimQA(GB,3); 37

In order to express the last GB element, viz. gap> PrintNP(GB[Length(GB)]); cabcabca + cbacba - ca as a combination of elements of KI, we use : gap> PrintTracePol(GBT[Length(GBT)]); - G(9)bacba + cG(10)

We compute matrices for left multiplication by generators using and determine the minimal polynomial of the sum of the three matrices.

gap> B := BaseQA(GB,3,0);; gap> M := MatricesQA(3,B,GB);; gap> f := MinimalPolynomial(Rationals,M[1]+M[2]+M[3]); x_1^7-6*x_1^5+9*x_1^3-3*x_1 gap> Factors(f); [ x_1, x_1^6-6*x_1^4+9*x_1^2-3 ]

It turns out that there are three non-zero numbers u,v,w such that the eigenvalues of the sum are 0,u,v,w,-u,-v,-w. This is the information used in .
<#/GAPDoc>