# Difference between revisions of "ApCoCoA-1:Weyl.AnnFs"

## Weyl.AnnFs

Computes annihilating ideal of a polynomial F^s in Weyl algebra A_n.

### Syntax

```Weyl.AnnFs(F:POLY):IDEAL
```

### Description

Please note: The function(s) explained on this page is/are using the ApCoCoAServer. You will have to start the ApCoCoAServer in order to use it/them.

This function computes annihilating ideal of a polynomial F^s using the Algorithm of Oaku and Takayama, where F is a polynomial in Weyl algebra A_n. F should not involve any of the indeterminates in {y1, ..., yn}. This ideal belongs to the Weyl algebra A_s =D[s]= QQ[x1, ..., xn, y1, ..., yn, s,w] where s commutes with all x_i and y_i's and w is redundant indeterminate used just to create internal structure of the weyl algebra.

• @param F A polynomial F in the indeterminates x1, ..., xn of a Weyl Algebra A_n.

• @return An ideal in A_s=QQ[x1, ..., xn,y1, ...,yn, s,w].

#### Example

```A2::=QQ[x[1..2],d[1..2]]; --Define appropriate ring
Use A2;
F:=x^3-x^2;
-------------------------------
AnnI:=Weyl.AnnFs(F);
-- CoCoAServer: computing Cpu Time = 0.078
-------------------------------
Ideal of Ring A_s = QQ[x[1..2],y[1..2],s,w]
Where current indeterminates are mapped into ring A_s as follows:
x --> x and d --> y
x --> x and d --> y

-------------------------------
AnnI;
A_s :: Ideal(
3x^2y + 2xy,
2xy + 3xy - 6s)   --AnnI belongs to the new ring A_s
-------------------------------
```

#### Example

```A3::=QQ[x[1..3],d[1..3]]; --Define appropriate ring
Use A3;

F:=x^2-xx-1;
Weyl.AnnFs(F);
-- CoCoAServer: computing Cpu Time = 0.14
-------------------------------
Ideal of Ring A_s = QQ[x[1..3],y[1..3],s,w]
A_s :: Ideal(2xy + xy, xy - xy, xxy + 2x^2y - 2xs + 2y, x^2y +
2xxy - 2xs - y, -x^2y + xxy - xs + y, xy + 2xy)
-------------------------------
```