# ApCoCoA-1:BBSGen.NonTriv

## BBSGen.NonTriv

This function computes the non-trivial polynomials of the generating set of the vanishing ideal of a border basis scheme.

### Syntax

```BBSGen.NonTriv(OO,BO,W,N);
BBSGen.NonTriv(OO:LIST,BO:LIST,W:MATRIX,N:INT):LIST;
```

### Description

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

• @return List of non-trivial generators of the vanishing ideal of the border basis scheme.

#### 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]];

Set Indentation;

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

```