Difference between revisions of "ApCoCoA-1:NC.MB"
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<short_description> | <short_description> | ||
Enumerate a Macaulay basis of a <tt>K</tt>-algebra. | Enumerate a Macaulay basis of a <tt>K</tt>-algebra. | ||
| + | <par/> | ||
| + | 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>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>). | ||
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| − | NC.MB( | + | NC.MB(G: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 | + | 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 (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions. |
<itemize> | <itemize> | ||
| − | <item>@param <em> | + | <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> | + | <item>@return: a LIST of words forming a Macaulay basis of the K-algebra <tt>P/<G></tt>.</item> |
| − | + | </itemize> | |
| + | Optional parameter: | ||
| + | <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> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| Line 31: | Line 35: | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| + | <see>Use</see> | ||
| + | <see>NC.HF</see> | ||
| + | <see>NC.IsGB</see> | ||
<see>NC.SetOrdering</see> | <see>NC.SetOrdering</see> | ||
<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||
| Line 36: | Line 43: | ||
<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 13:01, 29 April 2013
NC.MB
Enumerate a Macaulay basis of a K-algebra.
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 LT{G}. Then the residue class of the words in B form a K-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 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 basis, it is necessary to set DB to a large enough INT in order to compute the whole Macaulay 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
