Difference between revisions of "ApCoCoA-1:BBSGen.NonTriv"

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{{Version|1}}
 
<command>
 
<command>
 
   <title>BBSGen.NonTriv</title>
 
   <title>BBSGen.NonTriv</title>
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   <short_description>: This function computes the non-trivial polynomials of the generating set \tau of the vanishing ideal of a border basis scheme.
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   <short_description>This function computes the non-trivial polynomials of the generating set of the vanishing ideal of a border basis scheme.
 
              
 
              
 
  </short_description>
 
  </short_description>
  
−
The functions  from the package borderbasis <ref>BB.Box</ref>,  <ref>BB.Border</ref> must be used as input.
 
 
    
 
    
 
<syntax>
 
<syntax>
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<itemize>
 
<itemize>
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   <item>@para Order ideal OO, border BO, the number of Indeterminates of the Polynomial Ring and the Weight Matrix.
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   <item>@para Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N] and the weight matrix(see <ref>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</ref>). </item>
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</item>
 
 
   <item>@return List of non-trivial generators of the vanishing ideal of the border basis scheme.  </item>
 
   <item>@return List of non-trivial generators of the vanishing ideal of the border basis scheme.  </item>
 
</itemize>
 
</itemize>
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Use R::=QQ[x[1..2]];
 
Use R::=QQ[x[1..2]];
  
−
OO:=BB.Box([1,1]);
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OO:=$apcocoa/borderbasis.Box([1,1]);
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BO:=BB.Border(OO);
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BO:=$apcocoa/borderbasis.Border(OO);
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W:=BBSGen.Wmat(OO,BO,N);
 
 
Mu:=Len(OO);
 
Mu:=Len(OO);
 
Nu:=Len(BO);
 
Nu:=Len(BO);
 
N:=Len(Indets());
 
N:=Len(Indets());
 +
W:=BBSGen.Wmat(OO,BO,N);
 
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
 
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]];  
  
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   <types>
 
   <types>
 
     <type>borderbasis</type>
 
     <type>borderbasis</type>
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     <type>ideal</type>
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  <type>bbsmingensys</type>
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     <type>list</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
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<see>BB.Border</see>
 
−
 
  
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<see>BBSGen.NonTriv</see>
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<see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see>
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<see>BBSGen.Poldeg</see>
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<see>ApCoCoA-1:BBSGen.Poldeg|BBSGen.Poldeg</see>
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  <key>Wmat</key>
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   <key>BBSGen.NonTriv</key>
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   <key>BBSGen.Wmat</key>
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<key>NonTriv</key>
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  <key>bbsmingensys.Wmat</key>
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−
   <wiki-category>Package_bbsmingensys</wiki-category>
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   <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category>
 
</command>
 
</command>

Latest revision as of 09:51, 7 October 2020

This article is about a function from ApCoCoA-1.

BBSGen.NonTriv

This function computes the non-trivial polynomials of the generating set of the vanishing ideal of a border basis scheme.



Syntax

BBSGen.NonTriv(OO,BO,W,N);
BBSGen.NonTriv(OO:LIST,BO:LIST,W:MATRIX,N:INT):LIST;

Description


  • @para Order ideal OO, border BO, the number of indeterminates of the polynomial ring K[x_1,...,x_N] and the weight matrix(see BBSGen.Wmat).

  • @return List of non-trivial generators of the vanishing ideal of the border basis scheme.


Example

Use R::=QQ[x[1..2]];

OO:=$apcocoa/borderbasis.Box([1,1]);
BO:=$apcocoa/borderbasis.Border(OO);
Mu:=Len(OO);
Nu:=Len(BO);
N:=Len(Indets());
W:=BBSGen.Wmat(OO,BO,N);
Use XX::=QQ[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]]; 

Set Indentation;

BBSGen.NonTriv(OO,BO,W,N);
[
  [
    t[1,2,1,2],
    c[1,2]c[3,1] + c[1,4]c[4,1] - c[1,3],
    R :: Vector(1, 2)],
  [
    t[1,2,1,3],
    -c[1,1]c[2,2] - c[1,3]c[4,2] + c[1,4],
    R :: Vector(2, 1)],
  [
    t[1,2,1,4],
    -c[1,1]c[2,4] + c[1,2]c[3,3] + c[1,4]c[4,3] - c[1,3]c[4,4],
    R :: Vector(2, 2)],
  [
    t[1,2,2,2],
    c[2,2]c[3,1] + c[2,4]c[4,1] - c[2,3],
    R :: Vector(1, 1)],
  [
    t[1,2,2,3],
    -c[2,1]c[2,2] - c[2,3]c[4,2] - c[1,2] + c[2,4],
    R :: Vector(2, 0)],
  [
    t[1,2,2,4],
    -c[2,1]c[2,4] + c[2,2]c[3,3] + c[2,4]c[4,3] - c[2,3]c[4,4] - c[1,4],
    R :: Vector(2, 1)],
  [
    t[1,2,3,2],
    c[3,1]c[3,2] + c[3,4]c[4,1] + c[1,1] - c[3,3],
    R :: Vector(0, 2)],
  [
    t[1,2,3,3],
    -c[2,2]c[3,1] - c[3,3]c[4,2] + c[3,4],
    R :: Vector(1, 1)],
  [
    t[1,2,3,4],
    -c[2,4]c[3,1] + c[3,2]c[3,3] + c[3,4]c[4,3] - c[3,3]c[4,4] + c[1,3],
    R :: Vector(1, 2)],
  [
    t[1,2,4,2],
    c[3,1]c[4,2] + c[4,1]c[4,4] + c[2,1] - c[4,3],
    R :: Vector(0, 1)],
  [
    t[1,2,4,3],
    -c[2,2]c[4,1] - c[4,2]c[4,3] - c[3,2] + c[4,4],
    R :: Vector(1, 0)],
  [
    t[1,2,4,4],
    -c[2,4]c[4,1] + c[3,3]c[4,2] + c[2,3] - c[3,4],
    R :: Vector(1, 1)]]



BBSGen.Wmat

BBSGen.Poldeg



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