# ApCoCoA-1:BBSGen.NonStandPoly

## 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)]]

```