# Package sagbi/SB.SAGBI

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## SB.SAGBI

Computes a finite SAGBI-basis of a subalgebra if existing.

### Syntax

```SB.SAGBI(G:LIST of POLY):LIST of POLY
```

### Description

This function computes a finite SAGBI-basis of a subalgebra S generated by the polynomials of the list G, if a finite SAGBI-basis of S is existing. Then a list of polynomials is returned which form a SAGBI-basis of S. Otherwise the computation runs until it is interrupted.

• @param G A list of polynomials which generates a subalgebra.

• @return A list of polynomials which form a finite SAGBI-basis of the subalgebra generated by G.

#### Example

```Use QQ[x[1..3]];
S := SB.SAGBI([x[1]^2-x[3]^2,x[1]*x[2]+x[3]^2,x[2]^2-2*x[3]^2]);
indent(S);
-----------------------------------------------------------------------------
[
x[2]^2 -2*x[3]^2,
x[1]*x[2] +x[3]^2,
x[1]^2 -x[3]^2,
x[1]^2*x[3]^2 +x[1]*x[2]*x[3]^2 +(1/2)*x[2]^2*x[3]^2 +(-1/2)*x[3]^4
]
```