ApCoCoA-1:BBSGen.NonStandPoly: Difference between revisions

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New page: <command> <title>BBSGen.NonStandPoly</title> <short_description> Finds the non-standard polynomials of the ring <tt>K[c_{ij}]</tt> with respect to the arrow grading. </short_descriptio...
 
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{{Version|1}}
<command>
<command>
   <title>BBSGen.NonStandPoly</title>
   <title>BBSGen.NonStandPoly</title>
   <short_description> Finds the non-standard polynomials of the ring <tt>K[c_{ij}]</tt> with respect to the arrow grading. </short_description>
   <short_description>This function computes the non-standard polynomial  generators of the vanishing ideal of border basis
scheme with respect to the arrow grading.
           
</short_description>
    
    
<syntax>
<syntax>
BBSGen.NonStand(OO:LIST,BO:LIST,N:INT,W:MATRIX):LIST
BBSGen.NonStandPoly(OO,BO,W,N);
BBSGen.NonStandPoly(OO:LIST,BO:LIST,W:MATRIX,N:INTEGER):LIST  
 
</syntax>
</syntax>
   <description>
   <description>Let W be the weight matrix with respect to the arrow grading(see <ref>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</ref>).
Let tau^kl_ij be a polynomials from the generating set Tau of the vanishing ideal of border basis scheme. It  is called standard, if deg_W(tau^kl_ij) has exactly one  strictly positive component. If tau^kl_ij is not standard then it is called non-standard. This function computes such non-standard polynomials.
<itemize>
  <item>@param The order ideal OO, BO border of OO , the number of indeterminates of the polynomial ring K[x_1,...,x_N] and the weight matrix(<ref>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</ref>).
</item>
  <item>@return List of polynomials  and their degree with respect to the arrow grading.</item>
</itemize>


<itemize>
<item>@param <em>OO</em> A list of terms representing an order ideal.</item>
  <item>@param <em>BO</em> A list of terms representing the border.</item>
<item>@param <em>N</em> The number of elements of the Polynomial ring <tt>K[x_1,...x_n]</tt>.</item>
<item>@param <em>W</em> The weight matrix.</item>
   
   
  <item>@return A list of non-standard polynomials from <tt>BBS=K[c_{ij}]</tt> with their degree vectors from field <tt>K</tt>.</item>
</itemize>
<example>
<example>
Use R::=QQ[x[1..2]];
Use R::=QQ[x[1..2]];


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




NonStandPoly(OO,BO,W,N);
BBSGen.NonStandPoly(OO,BO,W,N);


   [  c[1,2]c[3,1] + c[1,4]c[4,1] - c[1,3],
   [  c[1,2]c[3,1] + c[1,4]c[4,1] - c[1,3],
Line 53: Line 53:




-------------------------------
</example>


</example>
   </description>
   </description>
   <types>
   <types>
     <type>bbsmingensys</type>
     <type>borderbasis</type>
    <type>list</type>
    <type>apcocoaserver</type>
   </types>
   </types>
  <see>BBSGen.Wmat</see>
 
<see>BBSGen.NonStand</see>
<see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see>
  <see>BB.Box</see>
<see>ApCoCoA-1:BBSGen.NonStand|BBSGen.NonStand</see>
   <see>BB.Border</see>
   <key>NonStandPoly</key>
<key>Wmat</key>
   <key>BBSGen.NonStandPoly</key>
   <key>BBSGen.NonStandPoly</key>
   <key>bbsmingensys.NonStandPoly</key>
   <key>bbsmingensys.NonStandPoly</key>
   <wiki-category>Package_bbsmingensys</wiki-category>
   <wiki-category>ApCoCoA-1:Package_bbsmingensys</wiki-category>
</command>
</command>

Latest revision as of 09:50, 7 October 2020

This article is about a function from ApCoCoA-1.

BBSGen.NonStandPoly

This function computes the non-standard polynomial generators of the vanishing ideal of border basis

scheme with respect to the arrow grading.


Syntax

BBSGen.NonStandPoly(OO,BO,W,N); 
BBSGen.NonStandPoly(OO:LIST,BO:LIST,W:MATRIX,N:INTEGER):LIST

Description

Let W be the weight matrix with respect to the arrow grading(see BBSGen.Wmat).

Let tau^kl_ij be a polynomials from the generating set Tau of the vanishing ideal of border basis scheme. It is called standard, if deg_W(tau^kl_ij) has exactly one strictly positive component. If tau^kl_ij is not standard then it is called non-standard. This function computes such non-standard polynomials.

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

  • @return List of polynomials and their degree with respect to the arrow grading.


Example

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

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


BBSGen.NonStandPoly(OO,BO,W,N);

  [  c[1,2]c[3,1] + c[1,4]c[4,1] - c[1,3],
    R :: Vector(1, 2)],
  [ c[1,1]c[2,2] + c[1,3]c[4,2] - c[1,4],
    R :: Vector(2, 1)],
  [ 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)],
  [c[2,2]c[3,1] + c[2,4]c[4,1] - c[2,3],
    R :: Vector(1, 1)],
  [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)],
  [c[2,2]c[3,1] + c[3,3]c[4,2] - c[3,4],
    R :: Vector(1, 1)],
  [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)],
  [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.NonStand



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