Difference between revisions of "ApCoCoA-1:NCo.IsGB"
Line 3: | Line 3: | ||
<short_description> | <short_description> | ||
Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis. | Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis. | ||
− | |||
− | |||
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
Line 10: | Line 8: | ||
</syntax> | </syntax> | ||
<description> | <description> | ||
+ | Note that, given an ideal <tt>I</tt> and a word ordering <tt>Ordering</tt>, a set of non-zero polynomials <tt>G</tt> is called a <em>Groebner basis</em> of <tt>I</tt> with respect to <tt>Ordering</tt> if the leading word set <tt>LW{G}</tt> generates the leading word ideal <tt>LW(I)</tt>. The function checks whether a given finite LIST of non-zero polynomials <tt>G</tt> is a Groebner basis by using the <tt>Buchberger Criterion</tt>, i.e. <tt>G</tt> is a Groebner basis if the S-polynomials of all obstructions have the zero normal remainder with respect to <tt>G</tt>. | ||
+ | <par/> | ||
<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/> | <par/> |
Revision as of 19:57, 14 May 2013
NCo.IsGB
Check whether a finite LIST of non-zero polynomials in a free monoid ring is a Groebner basis.
Syntax
NCo.IsGB(G:LIST):BOOL
Description
Note that, given an ideal I and a word ordering Ordering, a set of non-zero polynomials G is called a Groebner basis of I with respect to Ordering if the leading word set LW{G} generates the leading word ideal LW(I). The function checks whether a given finite LIST of non-zero polynomials G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if the S-polynomials of all obstructions have the zero normal remainder with respect to G.
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.
Please set ring environment coefficient field K, alphabet (or set of indeterminates) X and ordering via the functions NCo.SetFp, NCo.SetX and NCo.SetOrdering, respectively, before using this function. The default coefficient field is Q, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions.
@param G: a LIST of non-zero polynomials in K<X>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <X> and C is the coefficient of W. For example, the polynomial f=xy-y+1 is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].
@return: a BOOL, which is True if G is a Groebner basis with respect to the current word ordering and False otherwise.
Example
NCo.SetX(<quotes>xyt</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>]]; G := [F1, F2,F3,F4]; NCo.IsGB(G); -- LLEX ordering (default ordering) False ------------------------------- NCo.SetOrdering(<quotes>ELIM</quotes>); NCo.IsGB(G); False -------------------------------
See also