# ApCoCoA-1:NC.GB

## NC.GB

Computes a (partial) two-sided Groebner basis of finitely generated ideal (using Buchberger's procedure) over a free associative `K`-algebra.

### Syntax

NC.GB(Polynomials:LIST):LIST 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 cases we do not know whether there exists a finite Groebner basis. Instead 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 positive integer 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 positive integer which gives 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 positive integer 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>); 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]; NC.GB(Generators); -- over Q (default field), LLEX ordering (default ordering) [[[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>]], [[1, <quotes>ttyy</quotes>], [-1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [-1, <quotes>ttyx</quotes>]]] ------------------------------- NC.SetFp(); -- set default Fp=F2 NC.GB(Generators); -- over F2, LLEX ordering [[[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>]], [[1, <quotes>ttyy</quotes>], [1, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [1, <quotes>ttyx</quotes>]]] ------------------------------- NC.SetFp(3); NC.GB(Generators); -- over F3, LLEX ordering [[[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>]], [[1, <quotes>ttyy</quotes>], [2, <quotes>ttty</quotes>]], [[1, <quotes>tyyx</quotes>], [2, <quotes>ttyx</quotes>]]] -------------------------------

### See also