Difference between revisions of "ApCoCoA-1:NCo.IsGB"
From ApCoCoAWiki
m (replaced <quotes> tag by real quotes) |
|||
| (4 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| + | {{Version|1}} | ||
<command> | <command> | ||
<title>NCo.IsGB</title> | <title>NCo.IsGB</title> | ||
<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 9: | ||
</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/> | ||
| − | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>NCo.SetFp</ref>, <ref>NCo.SetX</ref> and <ref>NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ( | + | Please set ring environment <em>coefficient field</em> <tt> K</tt>, <em>alphabet</em> (or set of indeterminates) <tt>X</tt> and <em>ordering</em> via the functions <ref>ApCoCoA-1:NCo.SetFp|NCo.SetFp</ref>, <ref>ApCoCoA-1:NCo.SetX|NCo.SetX</ref> and <ref>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</ref>, respectively, before using this function. The default coefficient field is <tt>Q</tt>, and the default ordering is the length-lexicographic ordering ("LLEX"). For more information, please check the relevant functions. |
<itemize> | <itemize> | ||
| − | <item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1, | + | <item>@param <em>G</em>: a LIST of non-zero polynomials in <tt>K<X></tt>. Each polynomial is represented as a LIST of monomials, which are LISTs of the form [C, W] where W is a word in <tt><X></tt> and C is the coefficient of W. For example, the polynomial <tt>f=xy-y+1</tt> is represented as F:=[[1,"xy"], [-1, "y"], [1,""]].</item> |
<item>@return: a BOOL, which is True if <tt>G</tt> is a Groebner basis with respect to the current word ordering and False otherwise.</item> | <item>@return: a BOOL, which is True if <tt>G</tt> is a Groebner basis with respect to the current word ordering and False otherwise.</item> | ||
</itemize> | </itemize> | ||
<example> | <example> | ||
| − | NCo.SetX( | + | NCo.SetX("xyt"); |
| − | F1 := [[1, | + | F1 := [[1,"xx"], [-1,"yx"]]; |
| − | F2 := [[1, | + | F2 := [[1,"xy"], [-1,"ty"]]; |
| − | F3 := [[1, | + | F3 := [[1,"xt"], [-1, "tx"]]; |
| − | F4 := [[1, | + | F4 := [[1,"yt"], [-1, "ty"]]; |
G := [F1, F2,F3,F4]; | G := [F1, F2,F3,F4]; | ||
NCo.IsGB(G); -- LLEX ordering (default ordering) | NCo.IsGB(G); -- LLEX ordering (default ordering) | ||
| Line 28: | Line 29: | ||
False | False | ||
------------------------------- | ------------------------------- | ||
| − | NCo.SetOrdering( | + | NCo.SetOrdering("ELIM"); |
NCo.IsGB(G); | NCo.IsGB(G); | ||
| Line 36: | Line 37: | ||
</description> | </description> | ||
<seealso> | <seealso> | ||
| − | <see>NCo.GB</see> | + | <see>ApCoCoA-1:NCo.GB|NCo.GB</see> |
| − | <see>NCo.LW</see> | + | <see>ApCoCoA-1:NCo.LW|NCo.LW</see> |
| − | <see>NCo.ReducedGB</see> | + | <see>ApCoCoA-1:NCo.ReducedGB|NCo.ReducedGB</see> |
| − | <see>NCo.SetFp</see> | + | <see>ApCoCoA-1:NCo.SetFp|NCo.SetFp</see> |
| − | <see>NCo.SetOrdering</see> | + | <see>ApCoCoA-1:NCo.SetOrdering|NCo.SetOrdering</see> |
| − | <see>NCo.SetX</see> | + | <see>ApCoCoA-1:NCo.SetX|NCo.SetX</see> |
| − | <see>NCo.TruncatedGB</see> | + | <see>ApCoCoA-1:NCo.TruncatedGB|NCo.TruncatedGB</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
</seealso> | </seealso> | ||
<types> | <types> | ||
| Line 54: | Line 55: | ||
<key>NCo.IsGB</key> | <key>NCo.IsGB</key> | ||
<key>IsGB</key> | <key>IsGB</key> | ||
| − | <wiki-category>Package_gbmr</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_gbmr</wiki-category> |
</command> | </command> | ||
Latest revision as of 13:40, 29 October 2020
| This article is about a function from ApCoCoA-1. |
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("xyt");
F1 := [[1,"xx"], [-1,"yx"]];
F2 := [[1,"xy"], [-1,"ty"]];
F3 := [[1,"xt"], [-1, "tx"]];
F4 := [[1,"yt"], [-1, "ty"]];
G := [F1, F2,F3,F4];
NCo.IsGB(G); -- LLEX ordering (default ordering)
False
-------------------------------
NCo.SetOrdering("ELIM");
NCo.IsGB(G);
False
-------------------------------
See also
