<Heading>Using the packages <Package>GBNP</Package>
and <Package>NMO</Package></Heading>
This package deals with polynomials in noncommutative algebras
and to do so makes use of the noncommutative polynomial operations
provided by the <Package>GBNP</Package> <Cite Key='GBNP'/> package,
and orderings provided by the <Package>NMO</Package> package,
which is now included within <Package>GBNP</Package>.
In this chapter we remind users how to call some of these operations.
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:
<List>
<Item>
The first is a list <M>m</M> of monomials.
</Item>
<Item>
The second is a list <M>c</M> of coefficients of these monomials.
</Item>
</List>
The two lists have the same length.
The polynomial represented by the ordered pair <M>[m,c]</M> is
<M>\sum_i c_i m_i</M>.
A monomial is a list of positive integers.
They are interpreted as the indices of the variables.
So, if <M>k = [1,3,2,2,1]</M> and the variables are <M>x,y,z</M>
(in this order), then <M>k</M> represents the monomial <M>xzy^2x</M>.
There are various ways to print these,
but the default uses variables <M>a,b,c,\ldots</M>.
The zero polynomial is represented by <C>[[],[]]</C> and
the polynomial <M>1</M> is represented by <C>[[[]],[1]]</C>.
The algorithms are applicable for the algebra
<M>\mathbb{F}[x_1,x_2,\ldots,x_t]</M>
of noncommutative polynomials in <A>t</A> variables
over the field <M>\mathbb{F}</M>.
Accordingly, the list <M>c</M> should contain elements of
<M>\mathbb F</M>.
<P/>
<Index>NP2GP</Index> <Index>GP2NP</Index>
The &GBNP; functions <C>GP2NP</C> and <C>NP2GP</C>
convert a polynomial to NP format and back again.
Polynomials returned by <C>NP2GP</C> print with their coefficients
enclosed in brackets.
Polynomials may also be printed using the function <C>PrintNP</C>.
The function PrintNPList is used to print a list of NPs,
with one polynomial per line.
The function <C>CleanNP</C> is used to collect terms and reorder them.
The default ordering is first by degree and then lexicographically - <C>MonomialGrlexOrdering</C>.
Alternative orderings are available -
see section <Ref Sect="sec-orderings"/>.
<P/>
<Example>
<![CDATA[
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
]]>
</Example>
The &GBNP; package computes &Grob; bases using the function <C>SGrobner</C>.
In the example below the polynomials <M>\{p,q\}</M> define an ideal in
<M>\mathbb{Z}[a,b]</M> which has a three element &Grob; basis.
<Example>
<![CDATA[
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
]]>
</Example>
</Section>
<Section Label="sec-orderings">
<Heading>Orderings for monomials</Heading>
The three monomial orderings provided by the main &GAP; library are
<C>MonomialLexOrdering</C>, <C>MonomialGrlexOrdering</C>
and <C>MonomialGrevlexOrdering</C>.
The first of these is the default used by <Package>GBNP</Package>.
<P/>
The <Package>NMO</Package> package is now part of the package
<Package>GBNP</Package>.
It provides a choice of orderings on monomials, including
lexicographic and length-lexicographic ones.
<P/>
<Example>
<![CDATA[
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
]]>
</Example>
</Section>
</Chapter>
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