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

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<title>NC.MB</title> | <title>NC.MB</title> | ||

<short_description> | <short_description> | ||

− | Enumerate a Macaulay 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 finitely generated two-sided ideal <tt>I</tt> in a finitely generated non-commutative polynomial ring <tt>P</tt> over <tt>K</tt>, we can consider the <tt>K</tt>-algebra <tt>P/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>LT{G}</tt>. Then the residue class of the words in <tt>B</tt> form a <tt>K</tt>-basis of <tt>P/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NC.HF</ref>) of <tt>P/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>NC.SetOrdering</ref>). | + | Given a finitely generated two-sided ideal <tt>I</tt> in a finitely generated non-commutative polynomial ring <tt>P</tt> over <tt>K</tt>, we can consider the <tt>K</tt>-algebra <tt>P/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>LT{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>P/I</tt>. For the sake of computing the values of the Hilbert function (see <ref>NC.HF</ref>) of <tt>P/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>NC.SetOrdering</ref>). |

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

<syntax> | <syntax> | ||

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<itemize> | <itemize> | ||

<item>@param <em>G:</em> a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!</item> | <item>@param <em>G:</em> a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST []. <em>Warning:</em> users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!</item> | ||

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

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

Optional parameter: | Optional parameter: | ||

<itemize> | <itemize> | ||

− | <item>@param <em>DB:</em> a positive INT, which is a degree bound of the lengths of words. <em>Note that</em> we set <tt>DB=32</tt> by default. Thus, in the case that <tt>P/<G></tt> has a finite Macaulay basis, it is necessary to set <tt>DB</tt> to a large enough INT in order to compute the whole Macaulay basis.</item> | + | <item>@param <em>DB:</em> a positive INT, which is a degree bound of the lengths of words. <em>Note that</em> we set <tt>DB=32</tt> by default. Thus, in the case that <tt>P/<G></tt> has a finite Macaulay's basis, it is necessary to set <tt>DB</tt> to a large enough INT in order to compute the whole Macaulay's basis.</item> |

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

<example> | <example> |

## Revision as of 13:20, 29 April 2013

## NC.MB

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

Given a finitely generated two-sided ideal `I` in a finitely generated non-commutative polynomial ring `P` over `K`, we can consider the `K`-algebra `P/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 `LT{G}`. Then the residue class of the words in `B` form a `K`-basis, called *Macaulay's basis*, of `P/I`. For the sake of computing the values of the Hilbert function (see NC.HF) of `P/I`, in this function we require that `G` has to be a Groebner basis with respect to a length compatible word ordering (see NC.SetOrdering).

### Syntax

NC.MB(G: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 non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.

@param

*G:*a LIST of non-zero non-commutative polynomials, which form a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial`f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5`is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial`0`is represented as the empty LIST [].*Warning:*users should take responsibility to ensure that G is indeed a Groebner basis with respect to a length compatible word ordering!@return: a LIST of words forming a Macaulay's basis of the K-algebra

`P/<G>`.

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`P/<G>`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

NC.SetX(<quotes>xyzt</quotes>); NC.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>]]]; NC.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