Difference between revisions of "ApCoCoA-1:NC.HF"
| Line 4: | Line 4: | ||
Enumerate the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. | Enumerate the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. | ||
<par/> | <par/> | ||
| − | + | Let <tt>P</tt> be a finitely generated non-commutative polynomial ring over <tt>K</tt>, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>P</tt>. Then <tt>P/I</tt> is a finitely generated <tt>K</tt>-algebra. Moreover, for every integer <tt>i</tt>, we let <tt>F_{i}</tt> be the <tt>K</tt>-vector subspace generated by the words of length less than or equal to <tt>i</tt>. Clearly, the set <tt>{F_{i}}</tt> is a filtration of <tt>P</tt>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>P/I</tt>. The <em>Hilbert function</em> of <tt>K</tt>-algebra <tt>P/I</tt> is a map <tt>HF: N --> N</tt> defined by <tt>HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I))</tt>, i.e. <tt>HF(i)</tt> is equal to the number of words of length <tt>i</tt> in a Macaulay's basis (see <ref>NC.MB</ref>) of <tt>P/I</tt>. | |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| − | NC.HF( | + | NC.HF(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 non-negative integers, which are values of the Hilbert function of the K-algebra <tt>P<Gb></tt>.</item> |
| − | + | </itemize> | |
| + | Optional parameter: | ||
| + | <itemize> | ||
| + | <item>@param <em>DB:</em> a positive INT, which is a degree bound of the Hilbert function. <em>Note that</em> we set <tt>DB=32</tt> by default. Thus, in the case that the <tt>K</tt>-dimension of <tt>P<Gb></tt> is finite, it is necessary to set <tt>DB</tt> to a large enough INT in order to compute all the values of the Hilbert function.</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| Line 30: | Line 32: | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| + | <see>Use</see> | ||
| + | <see>NC.IsGB</see> | ||
| + | <see>NC.MB</see> | ||
<see>NC.SetOrdering</see> | <see>NC.SetOrdering</see> | ||
<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||
| Line 35: | Line 40: | ||
<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 13:33, 29 April 2013
NC.HF
Enumerate the values of the Hilbert function of a finitely generated K-algebra.
Let P be a finitely generated non-commutative polynomial ring over K, and let I be a finitely generated two-sided ideal in P. Then P/I is a finitely generated K-algebra. Moreover, for every integer i, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Clearly, the set {F_{i}} is a filtration of P. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of P/I. The Hilbert function of K-algebra P/I is a map HF: N --> N defined by HF(i)=dim(F_{i}/(F_{i} intersects I))-dim(F_{i-1}/(F_{i-1} intersects I)), i.e. HF(i) is equal to the number of words of length i in a Macaulay's basis (see NC.MB) of P/I.
Syntax
NC.HF(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 non-negative integers, which are values of the Hilbert function of the K-algebra P<Gb>.
Optional parameter:
@param DB: a positive INT, which is a degree bound of the Hilbert function. Note that we set DB=32 by default. Thus, in the case that the K-dimension of P<Gb> is finite, it is necessary to set DB to a large enough INT in order to compute all the values of the Hilbert function.
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.HF(Gb, 5); [1, 4, 12, 34, 100, 292] -------------------------------
See also
