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NCo.IsGB

Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis.
Syntax
          
NCo.IsGB(G:LIST):BOOL

          

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

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 ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

Example
NCo.SetX("xyt");  
F1 := [[1,"xx"], [-1,"yx"]];   
F2 := [[1,"xy"], [-1,"ty"]];  
F3 := [[1,"xt"], [-1, "tx"]];  
F4 := [[1,"yt"], [-1, "ty"]];  
G := [F1, F2,F3,F4]; 
NCo.IsGB(G); -- LLEX ordering (default ordering)

False
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NCo.SetOrdering("ELIM");
NCo.IsGB(G);

False
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See Also