Difference between revisions of "ApCoCoA-1:NC.HF"
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<title>NC.HF</title> | <title>NC.HF</title> | ||
<short_description> | <short_description> | ||
| − | Hilbert function of <tt>K</tt>-algebra. | + | Hilbert function of a <tt>K</tt>-algebra. |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| Line 11: | Line 11: | ||
<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 environment coefficient field <tt>K</tt>, alphabet (or indeterminates) <tt>X</tt> and ordering | + | Please set ring environment <em>coefficient field</em> <tt>K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NC.SetFp</ref>, <ref>NC.SetX</ref> and <ref>NC.SetOrdering</ref>, respectively, before calling the function. Default coefficient field is <tt>Q</tt>. Default ordering is length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant functions. |
<itemize> | <itemize> | ||
| − | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> which is a Groebner basis of | + | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> which is a Groebner basis (w.r.t. a length compatible admissible ordering, say <tt>Ordering</tt>) 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 <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <em>Warning:</em> users should take responsibility to make sure that Gb is indeed a Groebner basis w.r.t. <tt>Ordering</tt>! In the case that Gb is a partical Groebner basis, the function outputs pseudo values.</item> |
| − | <item>@param <em>DegreeBound:</em> (optional) a positive integer which is a degree bound of Hilbert funtion.</item> | + | <item>@param <em>DegreeBound:</em> (optional) a positive integer which is a degree bound of Hilbert funtion. <em>Note that</em> we set <tt>DegreeBound=32</tt> by default. Thus to compute all the values of the Hilbert function, it is necessary to set <tt>DegreeBound</tt> to a larger enough number.</item> |
| − | <item>@return: a LIST of non-negative integers, which is a list of | + | <item>@return: a LIST of non-negative integers, which is a list of values of the Hilbert function of the K-algebra <tt>K<X>/(Gb)</tt>.</item> |
</itemize> | </itemize> | ||
<example> | <example> | ||
Revision as of 11:55, 7 June 2012
NC.HF
Hilbert function of a K-algebra.
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. Default coefficient field is Q. 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 outputs 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
