# ApCoCoA-1:BBSGen.BBFinder

## BBSGen.BBFinder

Let LF be a list of indeterminates from the ring K[t[k,l,i,j] that is the subset of the ring XX:=K[c[1..Mu,1..Nu],t[1..N,1..N,1..Mu,1..Mu]] . This function finds the defining equations of border basis scheme and their degrees that correspond to the elements of the list LF.

### Syntax

```BBSGen.BBFinder(LF,OO,N,Poly);
BBSGen.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.

• @param List of t[k,l,i,j] , order ideal OO, border BO, N the number of indeterminates of the polynomial ring K[x_1,...,x_N] and the Weight Matrix(see BBSGen.Wmat).

• @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]]]]

```