Difference between revisions of "ApCoCoA-1:NCo.HF"
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Enumerate values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. | Enumerate values of the Hilbert function of a finitely generated <tt>K</tt>-algebra. | ||
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| + | Let <tt>K<X></tt> be a finitely generated non-commutative polynomial ring, and let <tt>I</tt> be a finitely generated two-sided ideal in <tt>K<X></tt>. Then <tt>K<X>/I</tt> is a finitely generated <tt>I</tt>-algebra. For every integer <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>. Further, the filtration <tt>{F_{i}}</tt> induces a filtration <tt>{F_{i}/(F_{i} intersects I)}</tt> of <tt>K<X>/I</tt>. The <em>Hilbert function</em> of <tt>K<X>/I</tt> is a map <tt>HF: N --></tt> N 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>NCo.MB</ref>) of <tt>K<X>/I</tt>. | ||
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<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/> | ||
Revision as of 19:56, 14 May 2013
NCo.HF
Enumerate values of the Hilbert function of a finitely generated K-algebra.
Syntax
NCo.HF(Gb:LIST[, DB:INT]):LIST
Description
Let K<X> be a finitely generated non-commutative polynomial ring, and let I be a finitely generated two-sided ideal in K<X>. Then K<X>/I is a finitely generated I-algebra. For every integer 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>. Further, the filtration {F_{i}} induces a filtration {F_{i}/(F_{i} intersects I)} of K<X>/I. The Hilbert function of K<X>/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 NCo.MB) of K<X>/I.
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 NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before calling this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
@param Gb: a LIST of non-zero polynomials in K<X> forming a Groebner basis with respect to a length compatible word ordering. Each polynomial is represented as a LIST of monomials, which are LISTs 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 with respect to a length compatible word ordering! In the case that Gb is a partical Groebner basis, the function enumerates the values of a pseudo Hilbert function.
@return: a LIST of non-negative integers, which are values of the Hilbert function of the K-algebra K<X>/<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 K<X>/<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
NCo.SetX(<quotes>xyzt</quotes>); NCo.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>]]]; NCo.HF(Gb, 5); [1, 4, 12, 34, 100, 292] -------------------------------
See also
