# CoCoA:Elim

## Elim

eliminate variables

### Description

This function returns the ideal or module obtained by eliminating the

indeterminates X from M. The coefficient ring needs to be a field.

As opposed to this function, there is also the *modifier*, Elim,

used when constructing a ring (see <ttref>Orderings</ttref> and

#### Example

```  Use R ::= Q[t,x,y,z];
Set Indentation;
Elim(t,Ideal(t^15+t^6+t-x,t^5-y,t^3-z));
Ideal(
-z^5 + y^3,
-y^4 - yz^2 + xy - z^2,
-xy^3z - y^2z^3 - xz^3 + x^2z - y^2 - y,
-y^2z^4 - x^2y^3 - xy^2z^2 - yz^4 - x^2z^2 + x^3 - y^2z - 2yz - z,
-y^3z^3 + xz^3 - y^3 - y^2)
-------------------------------
Use R ::= Q[t,s,x,y,z,w];
t..x;
[t, s, x]
-------------------------------
Elim(t..x,Ideal(t-x^2zw,x^2-t,y^2t-w)); -- Note the use of t..x.
Ideal(-zw^2 + w)
-------------------------------
Use R ::= Q[t[1..2],x[1..4]];
I := Ideal(x[1]-t[1]^4,x[2]-t[1]^2t[2],x[3]-t[1]t[2]^3,x[4]-t[2]^4);
t;
[t[1], t[2]]
-------------------------------
Elim(t,I);                         -- Note the use t.
Ideal(x[3]^4 - x[1]x[4]^3, x[2]^4 - x[1]^2x[4])
-------------------------------
```

### Syntax

```Elim(X:INDETS,M:IDEAL):IDEAL
Elim(X:INDETS,M:MODULE):MODULE

where X is an indeterminate or a list of indeterminates.
```

```   <type>groebner</type>
<type>groebner-basic</type>
<type>ideal</type>
<type>module</type>
```