Difference between revisions of "ApCoCoA-1:NC.MB"

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(New page: <command> <title>NC.MB</title> <short_description> Macauley basis of <tt>K</tt>-algebra with respect to Ordering. </short_description> <syntax> NC.MB(Gb:LIST):LIST NC.MB(Gb:LIST, DegreeBou...)
 
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<title>NC.MB</title>
 
<title>NC.MB</title>
 
<short_description>
 
<short_description>
Macauley basis of <tt>K</tt>-algebra with respect to Ordering.
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Macaulay basis of <tt>K</tt>-algebra with respect to Ordering.
 
</short_description>
 
</short_description>
 
<syntax>
 
<syntax>
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<itemize>
 
<itemize>
 
<item>@param <em>Gb:</em> a LIST of non-zero polynomials in <tt>K&lt;X&gt;</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&lt;X&gt;</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>
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<item>@param <em>DegreeBound:</em> (optional) a positive integer which is a degree bound of Hilbert function.</item>
<item>@return: a LIST of terms which form the Macauley basis of <tt>K&lt;X&gt;/(Gb)</tt> with respect to Ordering.</item>
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<item>@return: a LIST of terms which form the Macaulay basis of <tt>K&lt;X&gt;/(Gb)</tt> with respect to Ordering.</item>
 
</itemize>
 
</itemize>
 
<example>
 
<example>

Revision as of 14:59, 3 May 2011

NC.MB

Macaulay basis of K-algebra with respect to Ordering.

Syntax

NC.MB(Gb:LIST):LIST
NC.MB(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 function.

  • @return: a LIST of terms which form the Macaulay basis of K<X>/(Gb) with respect to Ordering.

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.MB(Gb,3);

[[<quotes></quotes>], [<quotes>t</quotes>, <quotes>z</quotes>, <quotes>y</quotes>, <quotes>x</quotes>], [<quotes>tt</quotes>, <quotes>tz</quotes>, <quotes>ty</quotes>, <quotes>tx</quotes>, <quotes>zt</quotes>, <quotes>zz</quotes>, <quotes>zy</quotes>, <quotes>zx</quotes>, <quotes>yz</quotes>, <quotes>yy</quotes>, <quotes>yx</quotes>, <quotes>xz</quotes>], 
[<quotes>ttt</quotes>, <quotes>ttz</quotes>, <quotes>tty</quotes>, <quotes>ttx</quotes>, <quotes>tzt</quotes>, <quotes>tzz</quotes>, <quotes>tzy</quotes>, <quotes>tzx</quotes>, <quotes>tyz</quotes>, <quotes>tyx</quotes>, <quotes>txz</quotes>, <quotes>ztt</quotes>, <quotes>ztz</quotes>, <quotes>zty</quotes>, <quotes>ztx</quotes>,
<quotes>zzt</quotes>, <quotes>zzz</quotes>, <quotes>zzy</quotes>, <quotes>zzx</quotes>, <quotes>zyz</quotes>, <quotes>zyy</quotes>, <quotes>zyx</quotes>, <quotes>zxz</quotes>, <quotes>yzt</quotes>, <quotes>yzz</quotes>, <quotes>yzy</quotes>, <quotes>yzx</quotes>, <quotes>yyz</quotes>, <quotes>yyy</quotes>,
 <quotes>yxz</quotes>, <quotes>xzt</quotes>, <quotes>xzz</quotes>, <quotes>xzy</quotes>, <quotes>xzx</quotes>]]
-------------------------------

See also

NC.Add

NC.Deg

NC.FindPolynomials

NC.GB

NC.HF

NC.Interreduction

NC.Intersection

NC.IsGB

NC.KernelOfHomomorphism

NC.LC

NC.LT

NC.LTIdeal

NC.MB

NC.MinimalPolynomial

NC.Multiply

NC.NR

NC.ReducedGB

NC.SetFp

NC.SetOrdering

NC.SetRelations

NC.SetRules

NC.SetX

NC.Subtract

NC.UnsetFp

NC.UnsetOrdering

NC.UnsetRelations

NC.UnsetRules

NC.UnsetX

Introduction to CoCoAServer