ApCoCoA-1:BBSGen.BBFinder

BBSGen.BBFinder

We let the indeterminate t[k,l,i,j] represent the (i,j) ^th entry of matrix the operation [A_k,A_l] . Let LF be a list of such indeterminates from the ring XX. This function finds the polynomial in the position (i,j) of the matrix [A_k,A_l] and its degree which corresponds to the elements given in the list LF.

Syntax

BBFinder(LF,OO,N,Poly); 
BBFinder(LF:LIST,OO:LIST,BO:LIST,N:INTEGER,W:MATRIX):LIST

Description

In order to use this function, one should define the ring XX as given in the example.

This function may not work properly for bigger order ideals and rings with more than three indeterminates, since the indeterminates of the ring XX also grows rapidly with respect to them.

The functions <commandref>BB.Border</commandref> and <commandref>BB.Box</commandref> are from the package borderbasis.

• @param List of t[k,l,i,j] , 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. (see BBSGen.Wmat)

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]]; 

BBSGen.BBFinder([t[1,2,3,4],t[1,2,2,4]],OO,BO,N,W);

[ [   [   R :: Vector(1, 2)],
    [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(2, 1)],
    [  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]]]]
    

BBSGen.Wmat

BBSGen.NonTriv

BBSGen.Poldeg