# Documentation

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# `polylib`::`representByElemSym`

Represent symmetric by elementary symmetric polynomials

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## Syntax

```polylib::representByElemSym(`f`, `l`)
```

## Description

`polylib::representByElemSym(f, [x1,...,xn])` returns a polynomial `g` in the identifiers x1 through xn such that replacing each `xi` by the `i`-th elementary symmetric polynomial gives `f`.

The list `l` must have as many operands as `f` has indeterminates.

The result is `FAIL` if the input is not symmetric.

## Examples

### Example 1

The symmetric polynomial x2 + y2 can be written as (x + y)2 - 2 (xy):

`polylib::representByElemSym(poly(x^2+y^2), [u,v]);`

### Example 2

`polylib::representByElemSym` works over domains also:

```f:=poly(x^2+y^2, Dom::IntegerMod(7)): polylib::representByElemSym(f, [u,v])```

## Parameters

 `f` Symmetric polynomial `l` List of indeterminates

## Return Values

Result is a polynomial having the same coefficient ring as `f`.

## Algorithms

It is a well-known theorem that every symmetric polynomial can be written in this way.