This package deals with polynomials in noncommutative algebras and to do so makes use of the noncommutative polynomial operations provided by the GBNP [CK24] 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.
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:
The first is a list \(m\) of monomials.
The second is a list \(c\) of coefficients of these monomials.
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 [[],[]] and the polynomial \(1\) is represented by [[[]],[1]]. The algorithms are applicable for the algebra \(\mathbb{F}[x_1,x_2,\ldots,x_t]\) of noncommutative polynomials in t variables over the field \(\mathbb{F}\). Accordingly, the list \(c\) should contain elements of \(\mathbb F\).
The GBNP functions GP2NP and NP2GP convert a polynomial to NP format and back again. Polynomials returned by NP2GP print with their coefficients enclosed in brackets. Polynomials may also be printed using the function PrintNP. The function PrintNPList is used to print a list of NPs, with one polynomial per line. The function CleanNP is used to collect terms and reorder them. The default ordering is first by degree and then lexicographically - MonomialGrlexOrdering. Alternative orderings are available - see section 2.3.
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
The GBNP package computes Gröbner bases using the function SGrobner. In the example below the polynomials \(\{p,q\}\) define an ideal in \(\mathbb{Z}[a,b]\) which has a three element Gröbner basis.
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
The three monomial orderings provided by the main GAP library are MonomialLexOrdering, MonomialGrlexOrdering and MonomialGrevlexOrdering. 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.
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
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