### The Maxima on-line user's manual

Algebra Calculator

#### Elem

Function: elem (<ele>, <sym>, <lvar>) decomposes the symmetric polynomial <sym>, in the variables contained in the list <lvar>, in terms of the elementary symmetric functions given in the list <ele>. If the first element of <ele> is given, it will be the size of the alphabet, otherwise the size will be the degree of the polynomial <sym>. If values are missing in the list <ele>, formal values of the type <e1>, <e2>, etc. will be added. The polynomial <sym> may be given in three different forms: contracted (`elem` should then be 1, its default value), partitioned (`elem` should be 3), or extended (i.e. the entire polynomial, and `elem` should then be 2). The function `pui` is used in the same way.

On an alphabet of size 3 with <e1>, the first elementary symmetric function, with value 7, the symmetric polynomial in 3 variables whose contracted form (which here depends on only two of its variables) is <x^4-2*x*y> decomposes as follows in elementary symmetric functions:

```          (%i1) elem ([3, 7], x^4 - 2*x*y, [x, y]);
(%o1) 7 (e3 - 7 e2 + 7 (49 - e2)) + 21 e3```

```                                                   + (- 2 (49 - e2) - 2) e2
(%i2) ratsimp (%);
2
(%o2)             28 e3 + 2 e2  - 198 e2 + 2401```

Other functions for changing bases: `comp2ele`.

There are also some inexact matches for `elem`. Try `?? elem` to see them.

```(%o1)                                true
(%i2) ```

### Related Examples

##### elem-ratsimp

elem ([1,3], x^2*y+x^...

ratsimp (%);

Calculate

##### elem-multi_elem

multi_elem ([[3,e1,e2...

Calculate

##### elem-expand-tcontract

g2:(v-(x1+2*x2+3*x3))...

e:elem([2],tcontract...

e,e1=-x1,e2=x^2+1;

Calculate

##### elem-multi_elem-ratsimp

multi_elem ([[2, e1, ...

multi_elem ([[2, ex1,...

ratsimp(%);

Calculate

##### elem-ratsimp

elem ([3],e1 , [x,1/x]);

ratsimp (%);

Calculate

##### elem

elem ([1], x^2+y^2, [...

Calculate

w:2;

v1:x1+w*x2+w^2*x3;

v2:x2+w*x3+w^2*x1;

Calculate

##### elem-ratsimp

elem ([x,1/x],e1 , [x...

ratsimp (%);

Calculate

w;

v1:x1+w*x2+w^2*x3;

v2:x2+w*x3+w^2*x1;

Calculate

##### elem

elem ([1,2,3], x^3+y^...

Calculate