Difference between revisions of "ApCoCoA-1:FGLM.FGLM"
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| + | {{Version|1}} | ||
<command> | <command> | ||
<title>FGLM.FGLM</title> | <title>FGLM.FGLM</title> | ||
| − | <short_description> | + | <short_description>Performs a FGLM Groebner Basis conversion using ApCoCoAServer.</short_description> |
<syntax> | <syntax> | ||
| − | FGLM( | + | FGLM(GBInput:LIST, M:MAT):LIST |
| − | FGLM( | + | FGLM(GBInput:LIST):LIST |
</syntax> | </syntax> | ||
<description> | <description> | ||
| Line 13: | Line 14: | ||
FGLM Groebner Basis conversion. Please note that the ideal generated by | FGLM Groebner Basis conversion. Please note that the ideal generated by | ||
the given Groebner Basis must be zero-dimensional. The Groebner | the given Groebner Basis must be zero-dimensional. The Groebner | ||
| − | Basis contained in list | + | Basis contained in list GBInput will be converted into a Groebner |
| − | Basis with respect to term ordering <ref>Ord</ref>(M), i.e. M must be a matrix | + | Basis with respect to term ordering <ref>ApCoCoA-1:Ord|Ord</ref>(M), i.e. M must be a matrix |
specifying a term ordering. If the parameter M is not specified, ApCoCoA | specifying a term ordering. If the parameter M is not specified, ApCoCoA | ||
| − | will assume M = <ref>Ord</ref>(). Please note that the resulting polynomials belong | + | will assume M = <ref>ApCoCoA-1:Ord|Ord</ref>(). Please note that the resulting polynomials belong |
| − | to a different ring than the ones in | + | to a different ring than the ones in GBInput. |
<par/> | <par/> | ||
The return value will be the transformed Groebner basis polynomials. | The return value will be the transformed Groebner basis polynomials. | ||
<itemize> | <itemize> | ||
| − | <item>@param <em> | + | <item>@param <em>GBInput</em> A Groebner basis of a zero-dimensional ideal.</item> |
| + | <item>@return A Groebner basis of the ideal generated by the polynomials of GBInput. The polynomials of the new Groebner basis will belong to the polynomial ring with term ordering specified by <tt>M</tt> or <ref>ApCoCoA-1:Ord|Ord</ref>() in case M is not given.</item> | ||
| + | </itemize> | ||
| + | The following parameter is optional. | ||
| + | <itemize> | ||
<item>@param <em>M</em> A matrix representing a term ordering.</item> | <item>@param <em>M</em> A matrix representing a term ordering.</item> | ||
| − | |||
</itemize> | </itemize> | ||
<example> | <example> | ||
Use QQ[x, y, z], DegRevLex; | Use QQ[x, y, z], DegRevLex; | ||
| − | + | GBInput := [z^4 -3z^3 - 4yz + 2z^2 - y + 2z - 2, yz^2 + 2yz - 2z^2 + 1, y^2 - 2yz + z^2 - z, x + y - z]; | |
M := LexMat(3); | M := LexMat(3); | ||
| − | GBNew := FGLM( | + | GBNew := FGLM.FGLM(GBInput, M); |
Use QQ[x, y, z], Ord(M); | Use QQ[x, y, z], Ord(M); | ||
-- New basis (Lex) | -- New basis (Lex) | ||
BringIn(GBNew); | BringIn(GBNew); | ||
| + | |||
| + | ------------------------------- | ||
| + | [z^6 - z^5 - 4z^4 - 2z^3 + 1, y - 4/7z^5 + 5/7z^4 + 13/7z^3 + 10/7z^2 - 6/7z - 2/7, | ||
| + | x + 4/7z^5 - 5/7z^4 - 13/7z^3 - 10/7z^2 - 1/7z + 2/7] | ||
| + | ------------------------------- | ||
</example> | </example> | ||
</description> | </description> | ||
| − | <see>GBasis5, and more</see> | + | <see>ApCoCoA-1:GBasis5, and more|GBasis5, and more</see> |
| − | <see>Introduction to CoCoAServer</see> | + | <see>ApCoCoA-1:Introduction to CoCoAServer|Introduction to CoCoAServer</see> |
<types> | <types> | ||
<type>groebner</type> | <type>groebner</type> | ||
| − | <type> | + | <type>ideal</type> |
<type>apcocoaserver</type> | <type>apcocoaserver</type> | ||
</types> | </types> | ||
| Line 46: | Line 55: | ||
<key>fglm.FGLM</key> | <key>fglm.FGLM</key> | ||
<key>groebner basis conversion</key> | <key>groebner basis conversion</key> | ||
| − | <wiki-category>Package_fglm</wiki-category> | + | <wiki-category>ApCoCoA-1:Package_fglm</wiki-category> |
</command> | </command> | ||
Latest revision as of 10:00, 7 October 2020
| This article is about a function from ApCoCoA-1. |
FGLM.FGLM
Performs a FGLM Groebner Basis conversion using ApCoCoAServer.
Syntax
FGLM(GBInput:LIST, M:MAT):LIST FGLM(GBInput:LIST):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.
The function FGLM calls the ApCoCoAServer to perform a
FGLM Groebner Basis conversion. Please note that the ideal generated by
the given Groebner Basis must be zero-dimensional. The Groebner Basis contained in list GBInput will be converted into a Groebner Basis with respect to term ordering Ord(M), i.e. M must be a matrix specifying a term ordering. If the parameter M is not specified, ApCoCoA will assume M = Ord(). Please note that the resulting polynomials belong to a different ring than the ones in GBInput.
The return value will be the transformed Groebner basis polynomials.
@param GBInput A Groebner basis of a zero-dimensional ideal.
@return A Groebner basis of the ideal generated by the polynomials of GBInput. The polynomials of the new Groebner basis will belong to the polynomial ring with term ordering specified by M or Ord() in case M is not given.
The following parameter is optional.
@param M A matrix representing a term ordering.
Example
Use QQ[x, y, z], DegRevLex; GBInput := [z^4 -3z^3 - 4yz + 2z^2 - y + 2z - 2, yz^2 + 2yz - 2z^2 + 1, y^2 - 2yz + z^2 - z, x + y - z]; M := LexMat(3); GBNew := FGLM.FGLM(GBInput, M); Use QQ[x, y, z], Ord(M); -- New basis (Lex) BringIn(GBNew); ------------------------------- [z^6 - z^5 - 4z^4 - 2z^3 + 1, y - 4/7z^5 + 5/7z^4 + 13/7z^3 + 10/7z^2 - 6/7z - 2/7, x + 4/7z^5 - 5/7z^4 - 13/7z^3 - 10/7z^2 - 1/7z + 2/7] -------------------------------
