Symmetric Function Theorem

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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 $f$.


Proof


Sources