Difference between revisions of "ApCoCoA-1:SB.IsSagbiOf"

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(New page: <command> <title>SB.IsSagbiOf</title> <short_description>Checks if a set of polynomials is a SAGBI-basis of a given subalgebra.</short_description> <syntax> SB.IsSagbiOf(Gens:LIST o...)
 
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{{Version|1|[[Package sagbi/SB.IsSAGBIOf]]}}
 
<command>
 
<command>
 
   <title>SB.IsSagbiOf</title>
 
   <title>SB.IsSagbiOf</title>
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</syntax>
 
</syntax>
 
   <description>
 
   <description>
This function if the given list of polynomials <tt>Basis</tt> is a SAGBI-basis
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This function checks if the given list of polynomials <tt>Basis</tt> forms a SAGBI-basis of the subalgebra <tt>S</tt> generated by the polynomials of the list <tt>Gens</tt>, i.e. it is checked if <tt>Basis</tt> is a SAGBI-Basis and if <tt>Basis</tt> also generates the subalgebra <tt>S</tt>. Then the corresponding boolean value is returned.
  
 
<itemize>
 
<itemize>
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SB.IsSagbiOf(G,SBasis);
 
SB.IsSagbiOf(G,SBasis);
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 +
-------------------------------------------------------
 +
-- output:
  
 
-- This SAGBI-basis generates the same subalgebra as the  
 
-- This SAGBI-basis generates the same subalgebra as the  
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SB.IsSagbiOf(G,SBasis);
 
SB.IsSagbiOf(G,SBasis);
 +
 +
-------------------------------------------------------
 +
-- output:
  
 
-- This SAGBI-basis generates the same subalgebra as the  
 
-- This SAGBI-basis generates the same subalgebra as the  
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SB.IsSagbiOf(G,Basis);
 
SB.IsSagbiOf(G,Basis);
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 +
-------------------------------------------------------
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-- output:
  
 
FALSE
 
FALSE
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</example>
 
</example>
 
   </description>
 
   </description>
<see>SB.Sagbi</see>
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<see>ApCoCoA-1:SB.Sagbi|SB.Sagbi</see>
<see>SB.IsSagbi</see>
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<see>ApCoCoA-1:SB.IsSagbi|SB.IsSagbi</see>
 
   <types>
 
   <types>
 
     <type>sagbi</type>
 
     <type>sagbi</type>
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   <key>sb.issagbiof</key>
 
   <key>sb.issagbiof</key>
 
   <key>sagbi.issagbiof</key>
 
   <key>sagbi.issagbiof</key>
   <wiki-category>Package_sagbi</wiki-category>
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   <wiki-category>ApCoCoA-1:Package_sagbi</wiki-category>
 
</command>
 
</command>

Latest revision as of 17:45, 27 October 2020

This article is about a function from ApCoCoA-1. If you are looking for the ApCoCoA-2 version of it, see Package sagbi/SB.IsSAGBIOf.

SB.IsSagbiOf

Checks if a set of polynomials is a SAGBI-basis of a given subalgebra.

Syntax

SB.IsSagbiOf(Gens:LIST of POLY, Basis:LIST of POLY):BOOL

Description

This function checks if the given list of polynomials Basis forms a SAGBI-basis of the subalgebra S generated by the polynomials of the list Gens, i.e. it is checked if Basis is a SAGBI-Basis and if Basis also generates the subalgebra S. Then the corresponding boolean value is returned.

  • @param Gens A list of polynomials, which are the generators of the current subalgebra.

  • @param Basis A list of polynomials, possibly a SAGBI-basis of the current subalgebra.

  • @return The corresponding boolean value.

Example

Set Indentation;
Use R::=QQ[x,y], DegLex;

G:=[x^2-y^2,x^2y,x^2y^2-y^4,x^2y^4,y^6x^2y^6-y^8];
SBasis:=SB.Sagbi(G);
SBasis;

SB.IsSagbiOf(G,SBasis);

-------------------------------------------------------
-- output:

-- This SAGBI-basis generates the same subalgebra as the 
-- the polynomials of the set G
[
  x^2 - y^2,
  x^2y,
  x^2y^2 - y^4,
  x^2y^4,
  x^2y^12 - y^8,
  y^6,
  x^2y^6 - y^8,
  x^2y^16 + x^4y^8 - y^12,
  x^2y^10 - 3/8y^12,
  y^14 - y^8,
  y^14 - y^8]
-------------------------------
TRUE
-------------------------------
-- Done.
-------------------------------

Example

Set Indentation;
Use R::=QQ[x,y], DegLex;

G:=[x+y,xy];
SBasis:=SB.Sagbi(G);
SBasis;

SB.IsSagbiOf(G,SBasis);

-------------------------------------------------------
-- output:

-- This SAGBI-basis generates the same subalgebra as the 
-- the polynomials of the set G
[
  x + y,
  xy]
-------------------------------
TRUE
-------------------------------
-- Done.
-------------------------------

Example

Set Indentation;
Use R::=QQ[x,y], DegLex;

G:=[x+y,xy];
Basis:=[x^3+x^2y];

-- The polynomial y^3+x^2y is not a member of K[G].
-- Therefore it is impossible that the given Basis generates
-- the same subalgebra.

SB.IsSagbiOf(G,Basis);

-------------------------------------------------------
-- output:

FALSE
-------------------------------
-- Done.
-------------------------------

SB.Sagbi

SB.IsSagbi