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

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+ | {{Version|1}} | ||

<command> | <command> | ||

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

<short_description> | <short_description> | ||

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. | ||

− | |||

− | |||

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

<syntax> | <syntax> | ||

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

<description> | <description> | ||

+ | Given a two-sided ideal <tt>I</tt> in a 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>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>P/I</tt>. For the sake of computing the values of the Hilbert-Dehn function (see <ref>ApCoCoA-1:NC.HF|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>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>). | ||

+ | <par/> | ||

<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 non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ( | + | Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions. |

<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> | ||

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

<example> | <example> | ||

− | + | Use ZZ/(2)[t,x,y]; | |

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

− | + | F1 := [[x^2], [y,x]]; -- x^2+yx | |

− | [[ | + | F2 := [[x,y], [t,y]]; -- xy+ty |

+ | F3 := [[x,t], [t,x]]; -- xt+tx | ||

+ | F4 := [[y,t], [t,y]]; -- yt+ty | ||

+ | G := [F1, F2,F3,F4]; | ||

+ | Gb:=NC.GB(G); | ||

NC.MB(Gb,3); | NC.MB(Gb,3); | ||

− | [[ | + | |

− | [ | + | [[[1]], [[y], [x], [t]], [[y, t], [y, x], [y^2], [x, t], [t^2]], |

− | + | [[y^3], [y^2, x], [y^2, t], [x, t^2], [t^3]]] | |

− | |||

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

</example> | </example> | ||

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

<seealso> | <seealso> | ||

− | <see>Use</see> | + | <see>ApCoCoA-1:Use|Use</see> |

− | <see>NC.HF</see> | + | <see>ApCoCoA-1:NC.HF|NC.HF</see> |

− | <see>NC.IsGB</see> | + | <see>ApCoCoA-1:NC.IsGB|NC.IsGB</see> |

− | <see>NC.LW</see> | + | <see>ApCoCoA-1:NC.LW|NC.LW</see> |

− | <see>NC.SetOrdering</see> | + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> |

− | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |

</seealso> | </seealso> | ||

<types> | <types> | ||

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<key>NC.MB</key> | <key>NC.MB</key> | ||

<key>MB</key> | <key>MB</key> | ||

− | <wiki-category>Package_ncpoly</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category> |

</command> | </command> |

## Latest revision as of 13:35, 29 October 2020

This article is about a function from ApCoCoA-1. |

## NC.MB

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

### Syntax

NC.MB(G:LIST[, DB:INT]):LIST

### Description

Given a two-sided ideal `I` in a 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 `LW{G}`. Then the residue class of the words in `B` form a `K`-basis, called a *Macaulay's basis*, of `P/I`. For the sake of computing the values of the Hilbert-Dehn 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).

*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

Use ZZ/(2)[t,x,y]; NC.SetOrdering("LLEX"); F1 := [[x^2], [y,x]]; -- x^2+yx F2 := [[x,y], [t,y]]; -- xy+ty F3 := [[x,t], [t,x]]; -- xt+tx F4 := [[y,t], [t,y]]; -- yt+ty G := [F1, F2,F3,F4]; Gb:=NC.GB(G); NC.MB(Gb,3); [[[1]], [[y], [x], [t]], [[y, t], [y, x], [y^2], [x, t], [t^2]], [[y^3], [y^2, x], [y^2, t], [x, t^2], [t^3]]] -------------------------------

### See also