BBTau

BBSGen.BBTau

Let Tau^kl_ij :=t[k,l,i,j] be the (i,j) ^th entry of matrix the operation [A_k,A_l].

For every k,l from {1,..,n} and i,j from {1,...,\mu}, this function finds the polynomial and its degree which corresponds to  Tau^kl_ij.

Syntax

BBTau(OO,BO,W,N);
BBTau(OO:LIST,BO:LIST,W:MATRIX,N:INT):LIST

Description

• @param The order ideal OO, border BO, the number of Indeterminates of the Polynomial Ring and the Weight Matrix.

• @return List of generators of the vanishing ideal of the border basis, their position in the matrix [A_k,A_l] and their degree wrt. arrow grading.

Example

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

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

Set Indentation; 

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

[ [t[1,2,1,1], 0,0],
[ 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[1,2,2,1],0,0],
  [  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[1,2,3,1], 0,0],
  [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[1,2,4,1],  0,0],
  [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)]]
-------------------------------

BB.Border

BB.Box

BBSGen.Wmat