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

(New page: <command> <title>NCo.MB</title> <short_description> Enumerate Macaulay basis of a <tt>K</tt>-algebra. </short_description> <syntax> NCo.MB(Gb:LIST):LIST NCo.MB(Gb:LIST, DegreeBound:INT):LI...) |
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<title>NCo.MB</title> | <title>NCo.MB</title> | ||

<short_description> | <short_description> | ||

− | Enumerate Macaulay basis of a <tt>K</tt>-algebra. | + | Enumerate a Macaulay's basis of a finitely generated <tt>K</tt>-algebra. |

+ | <par/> | ||

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

</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

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

− | |||

</syntax> | </syntax> | ||

<description> | <description> | ||

<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>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before | + | 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>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>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 (<quotes>LLEX</quotes>). 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> | + | <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,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <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>@param <em> | + | |

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

+ | </itemize> | ||

+ | Optional parameter: | ||

+ | <itemize> | ||

+ | <item>@param <em>DB:</em> 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 <tt>K<X>/<Gb></tt> 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.</item> | ||

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

<example> | <example> | ||

Line 31: | Line 36: | ||

</description> | </description> | ||

<seealso> | <seealso> | ||

+ | <see>NCo.HF</see> | ||

+ | <see>NCo.IsGB</see> | ||

<see>NCo.SetFp</see> | <see>NCo.SetFp</see> | ||

<see>NCo.SetOrdering</see> | <see>NCo.SetOrdering</see> | ||

Line 38: | Line 45: | ||

<types> | <types> | ||

<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||

+ | <type>ideal</type> | ||

<type>groebner</type> | <type>groebner</type> | ||

− | |||

<type>non_commutative</type> | <type>non_commutative</type> | ||

</types> | </types> |

## Revision as of 12:34, 30 April 2013

## NCo.MB

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

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 *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).

### Syntax

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

### Description

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

### See also