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

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<title>NC.GB</title> | <title>NC.GB</title> | ||

<short_description> | <short_description> | ||

− | Compute (partial) two-sided Groebner basis of finitely generated ideal ( | + | Compute (partial) two-sided Groebner basis of finitely generated ideal (using Buchberger's procedure) over a free associative <tt>K</tt>-algebra. |

</short_description> | </short_description> | ||

<syntax> | <syntax> | ||

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

<itemize> | <itemize> | ||

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

− | <item>@return: a | + | <item>@return: a list of polynomials, which is a reduced Groebner basis if a finite Groebner basis exists or a interreduced partial Groebner basis.</item> |

</itemize> | </itemize> | ||

− | About the optional parameters: For most | + | 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. |

<itemize> | <itemize> | ||

− | <item>@param <em>DegreeBound:</em> (optional) a | + | <item>@param <em>DegreeBound:</em> (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 <tt>S-element</tt> reaches <tt>DegreeBound</tt>, the function finishes the loop and returns a interreduced partial Groebner basis.</item> |

− | <item>@param <em>LoopBound:</em> (optional) a | + | <item>@param <em>LoopBound:</em> (optional) a positive integer which gives a limitation on the loop of Buchberger's procedure. When it runs through the main loop <tt>LoopBound</tt> times, the function stops the loop and returns a interreduced partial Groebner basis.</item> |

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

</itemize> | </itemize> | ||

<example> | <example> |

## Revision as of 13:57, 14 October 2010

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

### See also