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

From ApCoCoAWiki
m (insert version info)
 
(7 intermediate revisions by 3 users not shown)
Line 1: Line 1:
 +
{{Version|1}}
 
<command>
 
<command>
   <title>BBSGen.Wmat</title>
+
   <title>BBSGen.NonStandPoly</title>
 
   <short_description>This function computes the non-standard polynomial  generators of the vanishing ideal of border basis
 
   <short_description>This function computes the non-standard polynomial  generators of the vanishing ideal of border basis
 
scheme with respect to the arrow grading.
 
scheme with respect to the arrow grading.
Line 11: Line 12:
  
 
</syntax>
 
</syntax>
   <description>Let W be the weight matrix with respect to the arrow grading.(see <ref>BBSGen.Wmat</ref>)  
+
   <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 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.
+
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>
 
<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>BBSGen.Wmat</ref>).
+
   <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>
   <item>@return List of polynomials  and their degree with respect to the arrow grading.</item>
+
   <item>@return List of polynomials  and their degree with respect to the arrow grading.</item>
 
</itemize>
 
</itemize>
  
Line 57: Line 58:
 
   <types>
 
   <types>
 
     <type>borderbasis</type>
 
     <type>borderbasis</type>
     <type>ideal</type>
+
     <type>list</type>
 
     <type>apcocoaserver</type>
 
     <type>apcocoaserver</type>
 
   </types>
 
   </types>
  
  <see>BBSGen.Wmat</see>
+
  <see>ApCoCoA-1:BBSGen.Wmat|BBSGen.Wmat</see>
<see>BBSGen.NonStand</see>
+
<see>ApCoCoA-1:BBSGen.NonStand|BBSGen.NonStand</see>
   <key>Wmat</key>
+
   <key>NonStandPoly</key>
   <key>BBSGen.Wmat</key>
+
   <key>BBSGen.NonStandPoly</key>
   <key>bbsmingensys.Wmat</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