# Symmetric Function Theorem

## Theorem

Let $f$ be a polynomial in $n$ variables.

Let $f$ be of degree $r$ in each of its $n$ variables.

Then $f$ is equal to a polynomial of total degree $r$ with integer coefficients in the elementary symmetric functions:

$ds \sum x_i \sim x_i x_j, \dotsc, \prod x_j$

and the coefficients of $f$.