ApCoCoA-1:NCo.MB

This article is about a function from ApCoCoA-1.

NCo.MB

Enumerate a Macaulay's basis of a finitely generated K-algebra.

Syntax

NCo.MB(Gb:LIST[, DB:INT]):LIST

Description

Given a two-sided ideal I in a free monoid ring K<X>, we can consider the K-algebra K<X>/I as a K-vector space. Moreover, let G be a Groebner basis of I, and let B be the set of all words which are not a multiple of any word in the leading word set LW{G}. Then the residue class of the words in B form a K-basis, called a Macaulay's basis, of K<X>/I. For the sake of computing the values of the Hilbert function (see NCo.HF) of K<X>/I, in this function we require that G has to be a Groebner basis with respect to a length compatible word ordering (see NCo.SetOrdering).

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 Gb: a LIST of non-zero polynomials in K<X> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. Warning: users should take responsibility to make sure that Gb is indeed a Groebner basis with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates a pseudo basis.

• @return: a LIST of words forming a Macaulay's basis of the K-algebra K<X>/<Gb>.

Optional parameter:

• @param DB: a positive INT which is a degree bound of the lengths of words. Note that we set DB=32 by default. Thus, in the case that K<X>/<Gb> has a finite Macaulay's basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay's basis.

Example

NCo.SetX("xyzt");
NCo.SetOrdering("LLEX");
Gb:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]], 
[[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]];
NCo.MB(Gb,3);
[[""], ["t", "z", "y", "x"], ["tt", "tz", "ty", "tx", "zt", "zz", "zy", "zx", "yz", "yy", "yx", "xz"], 
["ttt", "ttz", "tty", "ttx", "tzt", "tzz", "tzy", "tzx", "tyz", "tyx", "txz", "ztt", "ztz", "zty", "ztx",
"zzt", "zzz", "zzy", "zzx", "zyz", "zyy", "zyx", "zxz", "yzt", "yzz", "yzy", "yzx", "yyz", "yyy",
 "yxz", "xzt", "xzz", "xzy", "xzx"]]
-------------------------------

See also

NCo.HF

NCo.IsGB

NCo.LW

NCo.SetFp

NCo.SetOrdering

NCo.SetX

Introduction to CoCoAServer