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<div class="ChapSects"><a href="chap2_mj.html#X86057870803DCA93">2 Using the packages GBNP
and NMO</a>
<div class="ContSect"> <a href="chap2_mj.html#X783C6EC87988B533">2.1 Noncommutative polynomials (NPs)</a>

</div>
<div class="ContSect"> <a href="chap2_mj.html#X7E4277497D877661">2.2 Gröbner Bases</a>

</div>
<div class="ContSect"> <a href="chap2_mj.html#X7F82A3608248CD31">2.3 Orderings for monomials</a>

</div>
</div>

<h3>2 Using the packages GBNP
and NMO</h3>

This package deals with polynomials in noncommutative algebras and to do so makes use of the noncommutative polynomial operations provided by the GBNP <a href="chapBib_mj.html#biBGBNP">[CK24]</a> package, and orderings provided by the NMO package, which is now included within GBNP. In this chapter we remind users how to call some of these operations.

<a id="X783C6EC87988B533" name="X783C6EC87988B533"></a>

<h4>2.1 Noncommutative polynomials (NPs)</h4>

Recall that the main datatype used by the GBNP package is a list of noncommutative polynomials (NPs). The data type for a noncommutative polynomial (its NP format) is a list of two lists:

<ul>
<li>The first is a list \(m\) of monomials.

</li>
<li>The second is a list \(c\) of coefficients of these monomials.

</li>
</ul>
The two lists have the same length. The polynomial represented by the ordered pair \([m,c]\) is \(\sum_i c_i m_i\). A monomial is a list of positive integers. They are interpreted as the indices of the variables. So, if \(k = [1,3,2,2,1]\) and the variables are \(x,y,z\) (in this order), then \(k\) represents the monomial \(xzy^2x\). There are various ways to print these, but the default uses variables \(a,b,c,\ldots\). The zero polynomial is represented by <code class="code">[[],[]]</code> and the polynomial \(1\) is represented by <code class="code">[[[]],[1]]</code>. The algorithms are applicable for the algebra \(\mathbb{F}[x_1,x_2,\ldots,x_t]\) of noncommutative polynomials in <var class="Arg">t</var> variables over the field \(\mathbb{F}\). Accordingly, the list \(c\) should contain elements of \(\mathbb F\).

The GBNP functions <code class="code">GP2NP</code> and <code class="code">NP2GP</code> convert a polynomial to NP format and back again. Polynomials returned by <code class="code">NP2GP</code> print with their coefficients enclosed in brackets. Polynomials may also be printed using the function <code class="code">PrintNP</code>. The function PrintNPList is used to print a list of NPs, with one polynomial per line. The function <code class="code">CleanNP</code> is used to collect terms and reorder them. The default ordering is first by degree and then lexicographically - <code class="code">MonomialGrlexOrdering</code>. Alternative orderings are available - see section <a href="chap2_mj.html#X7F82A3608248CD31">2.3</a>.

<div class="example"><pre>

gap> A3 := FreeAssociativeAlgebraWithOne(Rationals,"a","b","c");;
gap> a := A3.1;; b := A3.2;; c := A3.3;;
gap> ## define a polynomial and convert to NP-format
gap> p1 := 7*a^2*b*c + 8*b*c*a;
(8)*b*c*a+(7)*a^2*b*c
gap> Lp1 := GP2NP( p1 );
[ [ [ 1, 1, 2, 3 ], [ 2, 3, 1 ] ], [ 7, 8 ] ]
gap> ## define an NP-poly; clean it; and convert to a polynomial
gap> Lp2 := [ [ [1,1], [1,2,1], [3], [1,1], [3,1,2] ], [5,6,7,8,9] ];;
gap> PrintNP( Lp2 );
5a^2 + 6aba + 7c + 8a^2 + 9cab
gap> Lp2 := CleanNP( Lp2 );
[ [ [ 3, 1, 2 ], [ 1, 2, 1 ], [ 1, 1 ], [ 3 ] ], [ 9, 6, 13, 7 ] ]
gap> ## note the degree lexicographic ordering
gap> PrintNP( Lp2 );
9cab + 6aba + 13a^2 + 7c
gap> p2 := NP2GP( Lp2, A3 );
(9)*c*a*b+(6)*a*b*a+(13)*a^2+(7)*c
gap> PrintNPList( [ Lp1, Lp2, [ [], [] ], [ [ [] ], [9] ] ] );
7a^2bc + 8bca
9cab + 6aba + 13a^2 + 7c
0
9

</pre></div>

<a id="X7E4277497D877661" name="X7E4277497D877661"></a>

<h4>2.2 Gröbner Bases</h4>

The GBNP package computes Gröbner bases using the function <code class="code">SGrobner</code>. In the example below the polynomials \(\{p,q\}\) define an ideal in \(\mathbb{Z}[a,b]\) which has a three element Gröbner basis.

<div class="example"><pre>

gap> p := [ [ [2,2,2], [2,1], [1,2] ], [1,3,-1] ];;
gap> q := [ [ [1,1], [2] ], [1,1] ];; 
gap> PrintNPList( [p,q] );
b^3 + 3ba - ab
a^2 + b
gap> GB := SGrobner( [p,q] );;
gap> PrintNPList(GB);
a^2 + b
ba - ab
b^3 + 2ab

</pre></div>

<a id="X7F82A3608248CD31" name="X7F82A3608248CD31"></a>

<h4>2.3 Orderings for monomials</h4>

The three monomial orderings provided by the main GAP library are <code class="code">MonomialLexOrdering</code>, <code class="code">MonomialGrlexOrdering</code> and <code class="code">MonomialGrevlexOrdering</code>. The first of these is the default used by GBNP.

The NMO package is now part of the package GBNP. It provides a choice of orderings on monomials, including lexicographic and length-lexicographic ones.

<div class="example"><pre>

gap> Lp1;
[ [ [ 1, 1, 2, 3 ], [ 2, 3, 1 ] ], [ 7, 8 ] ]
gap> Lp2;
[ [ [ 3, 1, 2 ], [ 1, 2, 1 ], [ 1, 1 ], [ 3 ] ], [ 9, 6, 13, 7 ] ]
gap> GtNPoly( Lp1, Lp2 );
true
gap> ## select the lexicographic ordering and reorder p1, p2
gap> lexord := NCMonomialLeftLexicographicOrdering( A3 );;
gap> PatchGBNP( lexord );
LtNP patched.
GtNP patched.
gap> Lp1 := CleanNP( Lp1 );
[ [ [ 2, 3, 1 ], [ 1, 1, 2, 3 ] ], [ 8, 7 ] ]
gap> Lp2 := CleanNP( Lp2 );
[ [ [ 3, 1, 2 ], [ 3 ], [ 1, 2, 1 ], [ 1, 1 ] ], [ 9, 7, 6, 13 ] ]
gap> GtNPoly( Lp1, Lp2 );
false
gap> ## revert to degree lex order
gap> UnpatchGBNP();;
LtNP restored.
GtNP restored.
gap> Lp1 := CleanNP( Lp1 );
[ [ [ 1, 1, 2, 3 ], [ 2, 3, 1 ] ], [ 7, 8 ] ]
gap> Lp2 := CleanNP( Lp2 );
[ [ [ 3, 1, 2 ], [ 1, 2, 1 ], [ 1, 1 ], [ 3 ] ], [ 9, 6, 13, 7 ] ]
gap> GtNPoly( Lp1, Lp2 );
true

</pre></div>

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