# Difference between revisions of "ApCoCoA-1:NC.GB"

## NC.GB

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

### 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 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>]]]
-------------------------------
```