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Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis.
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.
- @param G: a LIST of non-zero polynomials in K. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].
- @return: a BOOL, which is True if G is a Groebner basis with respect to the current word ordering and False otherwise.
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
-------------------------------
NCo.SetOrdering("ELIM");
NCo.IsGB(G);
False
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