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Check whether a finite list (set) of non-zero polynomials in a finitely presented monoid ring is a Groebner basis.
Note that, given an ideal
I and an admissible ordering
Ordering, a set of non-zero polynomials
Gb is called a
Groebner basis of
I w.r.t.
Ordering if the leading term set
LT{Gb} (w.r.t.
Ordering) generates the leading term set
LT{I} (w.r.t.
Ordering) as a monoid ideal. The function check 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 all the
S-polynomials of obstructions have the zero normal remainder w.r.t.
G.
NC.MRIsGB(X:STRING, Ordering:STRING, Relations:LIST, G:LIST):BOOL
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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 coefficient field via the function
NC.SetFp (or
NC.UnsetFp) before calling this function. The default coefficient field is
Q. For more information, please check the relevant function.
- @param X: a finite alphabet or a finite set of indeterminates. It is of STRING type. Note that every letter in X MUST appear only once.The order of letters in X is very important, since it induces an admissible ordering.
- @param Ordering: an admissible ordering induced by the order of letters in X. It is represented as a STRING, which is a shortened-form of the name of corresponding admissible ordering. Note that "LLEX" (length-lexicographic ordering), "ELIM" (elimination ordering) and "LRLEX" (length-reverse-lexicographic ordering) are the only orderings supported currently. See NC.SetOrdering for more details.
- @param Relations: a finite set of relations. It is of LIST type. Each element in Relations is of the form [W1, W2], where W1 and W2 are words in . Each word is represented as a STRING. For example, the word xy^2x is represented as "xyyx", and the identity is represented as the empty string "". Thus, the relation (yx, xy) is represented as ["yx", "xy"], and the set of relations {(yx, xy),(zx,xz),(zy,yz)} is represented as [["yx", "xy"],["zx", "xz"],["zy", "yz"]].
- @param G: a LIST of non-zero polynomials in the finitely presented monoid ring. Each polynomial is represented as a LIST of monomials, which are pairs 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 value which is True if G is a Groebner basis (w.r.t. Ordering) and False otherwise.
X := "xyzt";
Ordering := "LLEX";
Relations:=[];
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];
NC.MRIsGB(X, Ordering, Relations, G);
False
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