Difference between revisions of "ApCoCoA-1:NC.HF"
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<title>NC.HF</title> | <title>NC.HF</title> | ||
<short_description> | <short_description> | ||
| − | + | Enumerate the values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. | |
<par/> | <par/> | ||
For every <tt>i</tt> in <tt>N</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>. Then <tt>{F_{i}}</tt> is a filtration of <tt>K<X></tt>. Let <tt>I</tt> be an ideal of <tt>K<X></tt>. The filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersect I)}</tt> of <tt>K<X>/I</tt>. The <em>Hilbert function</em> of <tt>K</tt>-algebra <tt>K<X>/I</tt> is a map from <tt>N</tt> to <tt>N</tt> defined by mapping <tt>i</tt> to <tt>dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I))</tt>. | For every <tt>i</tt> in <tt>N</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>. Then <tt>{F_{i}}</tt> is a filtration of <tt>K<X></tt>. Let <tt>I</tt> be an ideal of <tt>K<X></tt>. The filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersect I)}</tt> of <tt>K<X>/I</tt>. The <em>Hilbert function</em> of <tt>K</tt>-algebra <tt>K<X>/I</tt> is a map from <tt>N</tt> to <tt>N</tt> defined by mapping <tt>i</tt> to <tt>dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I))</tt>. | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
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<see>NC.SetOrdering</see> | <see>NC.SetOrdering</see> | ||
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<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||
</seealso> | </seealso> | ||
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<type>non_commutative</type> | <type>non_commutative</type> | ||
</types> | </types> | ||
| − | <key> | + | <key>ncpoly.HF</key> |
<key>NC.HF</key> | <key>NC.HF</key> | ||
<key>HF</key> | <key>HF</key> | ||
| − | <wiki-category> | + | <wiki-category>Package_ncpoly</wiki-category> |
</command> | </command> | ||
Revision as of 17:17, 25 April 2013
NC.HF
Enumerate the values of the Hilbert function of a finitely generated K-algebra.
For every i in N, we let F_{i} be the K-vector subspace generated by the words of length less than or equal to i. Then {F_{i}} is a filtration of K<X>. Let I be an ideal of K<X>. The filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersect I)} of K<X>/I. The Hilbert function of K-algebra K<X>/I is a map from N to N defined by mapping i to dim(F_{i}/(F_{i} intersect I))-dim(F_{i-1}/(F_{i-1} intersect I)).
Syntax
NC.HF(Gb:LIST):LIST NC.HF(Gb:LIST, DegreeBound: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 NC.SetFp, NC.SetX and NC.SetOrdering, respectively, before calling the function. The default coefficient field is Q. The default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
@param Gb: a LIST of non-zero polynomials in K<X> which is a Groebner basis (w.r.t. a length compatible admissible ordering, say Ordering) of the two-sided ideal generated by Gb. Each polynomial is represented as a LIST of monomials, which are pairs 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 w.r.t. Ordering! In the case that Gb is a partical Groebner basis, the function enumerates pseudo values.
@param DegreeBound: (optional) a positive integer which is a degree bound of Hilbert funtion. Note that we set DegreeBound=32 by default. Thus to compute all the values of the Hilbert function, it is necessary to set DegreeBound to a larger enough number.
@return: a LIST of non-negative integers, which is a list of values of the Hilbert function of the K-algebra K<X>/(Gb).
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
