ApCoCoA-1:NC.IsGB

NC.IsGB

Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.

Note that, given a word ordering, a set of non-zero polynomials G is called a Groebner basis of with respect to this ordering if the leading word set LW{G} generates the leading word ideal LW(<G>). This function checks whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if the S-polynomials of all obstructions of G have the zero normal remainder with respect to G.

Syntax

NC.IsGB(G:LIST):BOOL

Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

Please set non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

• @param G: a LIST of non-zero non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST [].

• @return: a BOOL, which is True if G is a Groebner basis with respect to the current ordering and False otherwise.

Example

Use ZZ/(2)[t,x,y,z];
G := [[[x^2], [y, x]], [[t, y], [x, y]], [[y, t], [x, y]], [[t, x], [x, t]], 
[[x, y, x], [y^2, x]], [[x, y^2], [y, x, y]], [[y, x, t], [y^2, x]]];
NC.SetOrdering("ELIM"); 
NC.IsGB(G);

True
-------------------------------
NC.SetOrdering("LLEX"); 
NC.IsGB(G);

False
-------------------------------

See also

Use

NC.GB

NC.LW

NC.RedGB

NC.SetOrdering

NC.TruncatedGB

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