# ApCoCoA-1:SB.IsInSubalgebra

## SB.IsInSubalgebra

Subalgebra membership test of a polynomial.

### Syntax

```SB.IsInSubalgebra(Gens:LIST of POLY, F:POLY):LIST
```

### Description

This functions checks the membership of the polynomial F in the subalgebra S generated by the polynomials of the list Gens. If F is not a member of S the function will return a list with the corresponding boolean value and NULL, otherwise TRUE and the subalgebra representation of F will be returned (see also SB.NFS and SB.SubalgebraPoly).

Important: This function works only, if a finite SAGBI-basis of S is existing!

• @param Gens A list of polynomials, which are the generators of the current subalgebra.

• @param F A polynomial.

• @return A list consisting of an boolean value and if it exists the subalgebra representation.

#### Example

```Use R::=QQ[x,y], DegLex;

F:=x^4+x^3y+x^2y^2+y^4;
G:=[x^2-y^2,x^2y,x^2y^2-y^4,x^2y^4,y^6x^2y^6-y^8];

SB.IsInSubalgebra(G,F);

-------------------------------------------------------
-- output:

[FALSE, NULL]
-------------------------------
-- Done.
-------------------------------
```

#### Example

```Use R::=QQ[x,y], DegLex;

F:=x^3+x^2y;
G:=[x+y,xy];

SB.IsInSubalgebra(G,F);

-------------------------------------------------------
-- output:

[FALSE, NULL]
-------------------------------
-- Done.
-------------------------------
```

#### Example

```Use R::=QQ[x,y], DegLex;

F:=x^4y^2+x^2y^4;
G:=[x^2-1,y^2-1];

L:=SB.IsInSubalgebra(G,F);
L;
SB.SubalgebraPoly(G,L);

-------------------------------------------------------
-- output:

[TRUE, [[2, 1, 1], [1, 2, 1], [2, 0, 1], [1, 1, 4], [0, 2, 1], [1, 0, 3], [0, 1, 3], [0, 0, 2]]]
-------------------------------
SARing :: y^2y + yy^2 + y^2 + 4yy + y^2 + 3y + 3y + 2
-------------------------------
-- Done.
-------------------------------
```