Difference between revisions of "ApCoCoA-1:NC.IsGB"
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Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring. | Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring. | ||
<par/> | <par/> | ||
| − | Note that, given | + | Note that, given a word ordering, a set of non-zero polynomials <tt>G</tt> is called a <em>Groebner basis</em> of with respect to this ordering if the leading word set <tt>LT{G}</tt> generates the leading word ideal <tt>LT(<G>)</tt>. This function checks whether a given finite set of non-zero polynomial <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 <tt>S-polynomials</tt> of all obstructions of <tt>G</tt> have the zero normal remainder with respect to <tt>G</tt>. |
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
| Line 12: | Line 12: | ||
<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/> | ||
| − | Please set ring | + | Please set non-commutative polynomial ring (via the command <ref>Use</ref>) and word ordering (via the function <ref>NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering (<quotes>LLEX</quotes>). For more information, please check the relevant commands and functions. |
<itemize> | <itemize> | ||
| − | <item>@param <em>G</em>: a LIST of non-zero polynomials | + | <item>@param <em>G</em>: a LIST of non-zero non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial <tt>f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5</tt> is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial <tt>0</tt> is represented as the empty LIST [].</item> |
| − | <item>@return: a BOOL | + | <item>@return: a BOOL, which is True if <tt>G</tt> is a Groebner basis with respect to the current ordering and False otherwise.</item> |
</itemize> | </itemize> | ||
<example> | <example> | ||
| Line 36: | Line 36: | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| + | <see>Use</see> | ||
<see>NC.GB</see> | <see>NC.GB</see> | ||
<see>NC.ReducedGB</see> | <see>NC.ReducedGB</see> | ||
Revision as of 12:43, 26 April 2013
NC.IsGB
Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.
Note that, given a word ordering, a set of non-zero polynomials G is called a Groebner basis of with respect to this ordering if the leading word set LT{G} generates the leading word ideal LT(<G>). This function checks whether a given finite set of non-zero polynomial G is a Groebner basis by using the Buchberger Criterion, i.e. G is a Groebner basis if the S-polynomials of all obstructions of G have the zero normal remainder with respect to G.
Syntax
NC.IsGB(G:LIST):BOOL
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.
Please set non-commutative polynomial ring (via the command Use) and word ordering (via the function NC.SetOrdering) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions.
@param G: a LIST of non-zero non-commutative polynomials. Each polynomial is represented as a LIST of LISTs, and each element in every inner LIST involves only one indeterminate or none (a constant). For example, the polynomial f=2x[2]y[1]x[2]^2-9y[2]x[1]^2x[2]^3+5 is represented as F:=[[2x[1],y[1],x[2]^2], [-9y[2],x[1]^2,x[2]^3], [5]]. The zero polynomial 0 is represented as the empty LIST [].
@return: a BOOL, which is True if G is a Groebner basis with respect to the current ordering and False otherwise.
Example
NC.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]; NC.IsGB(G); -- LLEX ordering (default ordering) False ------------------------------- NC.SetOrdering(<quotes>ELIM</quotes>); NC.IsGB(G); False -------------------------------
See also
