ApCoCoA-1:NC.GB

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Revision as of 21:04, 17 July 2010 by 132.231.183.6 (talk)

NC.GB

Compute (inter)reduced (partial) two-sided Groebner basis of finitely generated ideal (through Buchberger's procedure).

Syntax

NC.GB(Polynomials:LIST[, DegreeBound:INT, LoopBound:INT, Flag: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.

  • Before calling the function, please set ring environment coefficient field K, alphabet X and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(Ordering) respectively. Default coefficient field is Q. Default ordering is length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.

  • @param Polynomials: a LIST of polynomials generating a two-sided ideal in K<X>. Each polynomial in K<X> is represented as a LIST of LISTs, which are pairs of form [c, w] where c is in K and w is a word in X*. Unit in X* is empty word represented as an empty STRING "". 0 polynomial is represented as an empty LIST []. For example, polynomial F:=xy-y+1 in K<x,y> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].

  • @return: a LIST of polynomials, which is a reduced Groebner basis if a finite Groebner basis exists or a interreduced partial Groebner basis.

About the optional parameters.

  • For most of cases we don't know whether there exists a finite Groebner basis. In stead of forcing computer yelling and informing nothing valuable, the function has 3 optional parameters to control the computation. Note that at the moment all of the following 3 additional optional parameters must be used at the same time.

  • @param DegreeBound: (optional) a INT (natural number) which gives a limitation on the degree of polynomials during Buchberger's procedure. When the degree of normal remainder of some S-element reaches DegreeBound, the function finishes the loop and returns a interreduced partial Groebner basis.

  • @param LoopBound: (optional) a INT (natural number) which gives a a limitation on the loop of Buchberger's procedure. When it runs through the main loop LoopBound times, the function stops the loop and returns a interreduced partial Groebner basis.

  • @param Flag: (optional) a INT (natural number) which is a multi-switch for the output of ApCoCoAServer. If Flag=0, the server prints nothing on the screen. If Flag=1, the server prints basic information about computing procedure, such as number of S-elements has been checked and to be checked. If Flag=2, the server prints current partial Groebner basis before each loop as well. Note that the initial idea is to use Flag as a tool for debugging and tracing the computing process.

Example

NC.SetX(<quotes>xyzt</quotes>); 
NC.SetOrdering(<quotes>LLEX</quotes>); 
F1 := [[1,<quotes>xx</quotes>], [-1,<quotes>yx</quotes>]];   
F2 := [[1,<quotes>xy</quotes>], [-1,<quotes>ty</quotes>]];  
F3 := [[1,<quotes>xt</quotes>], [-1, <quotes>tx</quotes>]];  
F4 := [[1,<quotes>yt</quotes>], [-1, <quotes>ty</quotes>]];  
Generators := [F1, F2,F3,F4]; -- computation over Q
NC.GB(Generators);
[[[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.SetFp();
NC.GB(Generators); -- computation over F2
[[[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.SetFp(3);
NC.GB(Generators); -- computation over F3
[[[1, <quotes>yt</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xt</quotes>], [2, <quotes>tx</quotes>]], [[1, <quotes>xy</quotes>], [2, <quotes>ty</quotes>]], [[1, <quotes>xx</quotes>], [2, <quotes>yx</quotes>]], 
[[1, <quotes>tyy</quotes>], [2, <quotes>tty</quotes>]], [[1, <quotes>yyx</quotes>], [2, <quotes>tyx</quotes>]]]
-------------------------------

See also

NC.Add

NC.IsGB

NC.LC

NC.LT

NC.LTIdeal

NC.Multiply

NC.NR

NC.SetFp

NC.SetOrdering

NC.SetRelations

NC.SetRules

NC.SetX

NC.Subtract

NC.UnsetFp

NC.UnsetOrdering

NC.UnsetRelations

NC.UnsetRules

NC.UnsetX

Gbmr.MRSubtract

Gbmr.MRMultiply

Gbmr.MRBP

Gbmr.MRIntersection

Gbmr.MRKernelOfHomomorphism

Gbmr.MRMinimalPolynomials

Introduction to CoCoAServer