Difference between revisions of "ApCoCoA-1:NC.GB"
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<description> | <description> | ||
<em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | <em>Please note:</em> The function(s) explained on this page is/are using the <em>ApCoCoAServer</em>. You will have to start the ApCoCoAServer in order to use it/them. | ||
| + | <par/> | ||
| + | Before calling the function, please set ring environment coefficient field <tt>K</tt>, alphabet <tt>X</tt> and ordering through the functions NC.SetFp(Prime) (or NC.UnsetFp()), NC.SetX(X) and NC.SetOrdering(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 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 [c, w] 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>@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 [c, w] 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 LIST of polynomials, which is a reduced Groebner basis if a finite Groebner basis exists or a interreduced partial Groebner basis.</item> | <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> | ||
| Line 67: | Line 68: | ||
<see>NC.UnsetRules</see> | <see>NC.UnsetRules</see> | ||
<see>NC.UnsetX</see> | <see>NC.UnsetX</see> | ||
| − | <see> | + | <see>NC.MRAdd</see> |
| − | <see> | + | <see>NC.MRBP</see> |
| − | <see> | + | <see>NC.MRIntersection</see> |
| − | <see> | + | <see>NC.MRKernelOfHomomorphism</see> |
| − | <see> | + | <see>NC.MRMinimalPolynomials</see> |
| − | <see> | + | <see>NC.MRMultiply</see> |
| − | <see> | + | <see>NC.MRSubtract</see> |
<see>Introduction to CoCoAServer</see> | <see>Introduction to CoCoAServer</see> | ||
</seealso> | </seealso> | ||
| Line 80: | Line 81: | ||
<type>groebner</type> | <type>groebner</type> | ||
</types> | </types> | ||
| + | <key>gbmr.GB</key> | ||
<key>NC.GB</key> | <key>NC.GB</key> | ||
<key>GB</key> | <key>GB</key> | ||
<wiki-category>Package_gbmr</wiki-category> | <wiki-category>Package_gbmr</wiki-category> | ||
</command> | </command> | ||
Revision as of 10:29, 22 July 2010
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>); 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>]]] ------------------------------- 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>]]] ------------------------------- 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>]]] -------------------------------
See also
