# ApCoCoA-1:Weyl.WGB

## Weyl.WGB

Computes the Groebner basis of an ideal I in Weyl algebra A_n.

### Syntax

```Weyl.WGB(I:IDEAL):LIST
```

### 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 a Groebner Basis for an Ideal I = (f_1,f_2, ..., f_r) where every generator f_i should be a Weyl polynomial in Normal form.

• @param I An ideal in the Weyl algebra.

• @return A Groebner Basis of the given ideal.

#### Example

```A1::=QQ[x,d];	--Define appropraite ring
Use A1;
I:=Ideal(x,d);  -- Now start ApCoCoA server for executing next command
Weyl.WeylGB(I);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[1]
-------------------------------
Note that Groebner basis you obtained is minimal.
A2::=QQ[x[1..2],y[1..2]];
Use A2;
I1:=Ideal(x[1]^7,y[1]^7);
Weyl.WGB(I1);
-- CoCoAServer: computing Cpu Time = 0.094
-------------------------------
[1]
-------------------------------
```

#### Example

```W3::=ZZ/(7)[x[1..3],d[1..3]];
Use W3;
I2:=Ideal(x[1]^7,d[1]^7);  --is a 2-sided ideal in W3
Weyl.WGB(I2);   --ApCoCOAServer should be running
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[x[1]^7, d[1]^7]
-------------------------------

I3:=Ideal(x[1]^3d[2],x[2]*d[1]^2);

Weyl.WGB(I3);
-- CoCoAServer: computing Cpu Time = 0
-------------------------------
[x[2]^2d[2], x[2]d[2]^2 + 2d[2], x[1]^3d[1]^2 + x[1]^2x[2]d[1]d[2] + x[1]x[2]d[2], x[1]^3d[2], x[2]d[1]^2]
-------------------------------
```