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>Perform a FGLM Groebner Basis conversion using ApCoCoAServer.</short_description>
+
   <short_description>Performs a FGLM Groebner Basis conversion using ApCoCoAServer.</short_description>
   <syntax>FGLM(GBOld:LIST, M:MAT):LIST
+
    
FGLM(GBOld:LIST):LIST</syntax>
+
<syntax>
 +
FGLM(GBInput:LIST, M:MAT):LIST
 +
FGLM(GBInput:LIST):LIST
 +
</syntax>
 
   <description>
 
   <description>
 
<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.
Line 10: 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 GBOld will be converted into a Groebner
+
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 GBOld.
+
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>GBOld</em> A Groebner basis of a zero-dimensional ideal.</item>
+
   <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>
  <item>@return A Groebner basis of the ideal generated by the polynomials of GBOld. 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.</item>
 
 
</itemize>
 
</itemize>
 
<example>
 
<example>
 
Use QQ[x, y, z], DegRevLex;
 
Use QQ[x, y, z], DegRevLex;
GBOld := [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];
+
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(GBOld, M);
+
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>matrix</type>
+
     <type>ideal</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
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   <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]
-------------------------------

GBasis5, and more

Introduction to CoCoAServer