# Difference between revisions of "ApCoCoA-1:NCo.MB"

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

<par/> | <par/> | ||

− | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ( | + | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions. |

<itemize> | <itemize> | ||

− | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> 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 <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1, | + | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> 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 <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. <em>Warning:</em> 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.</item> |

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

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</itemize> | </itemize> | ||

<example> | <example> | ||

− | NCo.SetX( | + | NCo.SetX("xyzt"); |

− | NCo.SetOrdering( | + | NCo.SetOrdering("LLEX"); |

− | Gb:= [[[1, | + | Gb:= [[[1, "yt"], [-1, "ty"]], [[1, "xt"], [-1, "tx"]], [[1, "xy"], [-1, "ty"]], [[1, "xx"], [-1, "yx"]], |

− | [[1, | + | [[1, "tyy"], [-1, "tty"]], [[1, "yyx"], [-1, "tyx"]]]; |

NCo.MB(Gb,3); | 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"]] | |

------------------------------- | ------------------------------- | ||

</example> | </example> |

## Latest revision as of 13:41, 29 October 2020

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