Difference between revisions of "ApCoCoA-1:NC.HF"
(New page: <command> <title>NC.HF</title> <short_description> Hilbert function of <tt>K</tt>-algebra. </short_description> <syntax> NC.HF(Gb:LIST):LIST NC.HF(Gb:LIST, DegreeBound:INT):LIST </syntax> ...) |
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<item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> which is a Groebner basis of (two-sided) ideal generated by Gb. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</item> | <item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K<X></tt> which is a Groebner basis of (two-sided) ideal generated by Gb. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, <tt>xy^2x</tt> is represented as <quotes>xyyx</quotes>, unit is represented as an empty string <quotes></quotes>. Then, polynomial <tt>F=xy-y+1</tt> is represented as F:=[[1,<quotes>xy</quotes>], [-1, <quotes>y</quotes>], [1,<quotes></quotes>]]. <tt>0</tt> polynomial is represented as an empty LIST [].</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.</item> | ||
| − | <item>@return: a LIST of non-negative integers, which is a list of coefficient of Hilbert function of <tt>K<X></tt> | + | <item>@return: a LIST of non-negative integers, which is a list of coefficient of Hilbert function of <tt>K<X>/(Gb)</tt>.</item> |
</itemize> | </itemize> | ||
<example> | <example> | ||
Revision as of 20:19, 11 December 2010
NC.HF
Hilbert function of 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 indeterminates) X and ordering through the functions NC.SetFp(Prime), NC.SetX(X) and NC.SetOrdering(Ordering), 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 of (two-sided) ideal generated by Gb. Each polynomial is represented as a LIST of LISTs, which are pairs of form [C, W] where C is a coefficient and W is a word (or term). Each term is represented as a STRING. For example, xy^2x is represented as "xyyx", unit is represented as an empty string "". Then, polynomial F=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]]. 0 polynomial is represented as an empty LIST [].
@param DegreeBound: (optional) a positive integer which is a degree bound of Hilbert funtion.
@return: a LIST of non-negative integers, which is a list of coefficient of Hilbert function of K<X>/(Gb).
Example
NC.SetX(<quotes>abc</quotes>); NC.SetOrdering(<quotes>ELIM</quotes>); NC.RingEnv(); Coefficient ring : Q Alphabet : abc Ordering : ELIM ------------------------------- F1 := [[1,<quotes>a</quotes>],[1,<quotes></quotes>]]; F2 := [[1,<quotes>b</quotes>],[1,<quotes>ba</quotes>]]; NC.Add(F1,F2); -- over Q [[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]] ------------------------------- NC.SetFp(); -- set default Fp = F2 NC.RingEnv(); Coefficient ring : Fp = Z/(2) Alphabet : abc Ordering : ELIM ------------------------------- NC.Add(F1,F2); -- over F2 [[1, <quotes>ba</quotes>], [1, <quotes>a</quotes>], [1, <quotes>b</quotes>], [1, <quotes></quotes>]] ------------------------------- NC.Add(F1,F1); -- over F2 [ ] -------------------------------
See also
