Difference between revisions of "ApCoCoA-1:NC.IsGB"
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<command> | <command> | ||
<title>NC.IsGB</title> | <title>NC.IsGB</title> | ||
<short_description> | <short_description> | ||
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. | ||
| − | |||
| − | |||
</short_description> | </short_description> | ||
<syntax> | <syntax> | ||
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</syntax> | </syntax> | ||
<description> | <description> | ||
| + | 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>LW{G}</tt> generates the leading word ideal <tt>LW(<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 S-polynomials of all obstructions of <tt>G</tt> 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/> | ||
| − | 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 ( | + | Please set non-commutative polynomial ring (via the command <ref>ApCoCoA-1:Use|Use</ref>) and word ordering (via the function <ref>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</ref>) before calling this function. The default word ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant commands and functions. |
<itemize> | <itemize> | ||
<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>@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> | ||
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</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>Use</see> | + | <see>ApCoCoA-1:Use|Use</see> |
| − | <see>NC.GB</see> | + | <see>ApCoCoA-1:NC.GB|NC.GB</see> |
| − | <see>NC.LW</see> | + | <see>ApCoCoA-1:NC.LW|NC.LW</see> |
| − | <see>NC.RedGB</see> | + | <see>ApCoCoA-1:NC.RedGB|NC.RedGB</see> |
| − | <see>NC.SetOrdering</see> | + | <see>ApCoCoA-1:NC.SetOrdering|NC.SetOrdering</see> |
| − | <see>NC.TruncatedGB</see> | + | <see>ApCoCoA-1:NC.TruncatedGB|NC.TruncatedGB</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
| Line 51: | Line 52: | ||
<key>NC.IsGB</key> | <key>NC.IsGB</key> | ||
<key>IsGB</key> | <key>IsGB</key> | ||
| − | <wiki-category>Package_ncpoly</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_ncpoly</wiki-category> |
</command> | </command> | ||
Latest revision as of 13:34, 29 October 2020
| This article is about a function from ApCoCoA-1. |
NC.IsGB
Check whether a LIST of non-zero polynomials is a Groebner basis in a non-commutative polynomial ring.
Syntax
NC.IsGB(G:LIST):BOOL
Description
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 LW{G} generates the leading word ideal LW(<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.
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
Use ZZ/(2)[t,x,y];
G := [[[x^2], [y, x]], [[t, y], [x, y]], [[y, t], [x, y]], [[t, x], [x, t]],
[[x, y, x], [y^2, x]], [[x, y^2], [y, x, y]], [[y, x, t], [y^2, x]]];
NC.SetOrdering("ELIM");
NC.IsGB(G);
True
-------------------------------
NC.SetOrdering("LLEX");
NC.IsGB(G);
False
-------------------------------
See also
