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

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

<par/> | <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>). | + | 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 a <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> | ||

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<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>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>@return: a LIST of words forming a Macaulay's basis of the K-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> |

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

Optional parameter: | Optional parameter: | ||

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

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

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

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

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

## Revision as of 18:11, 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 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).

### 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