Symmetric Multivariate Real Polynomials
Definition
A polynomial is symmetric
if its value is unchanged under all permutations
of its arguments.
I.e., a polynomial $f: \R ^n \to \R $ is
symmetric if
\[
f(x) = f(x \circ g) \quad \text{for all permutations } g
\text{ of } \set{1, \dots , n}
\]
Example
The polynomial $f: \R ^n \to \R $ defined by
\[
f(x_1, \dots , x_n) = x_1 + x_2 + \cdots + x_n \quad
\text{for all } x_1, \dots , x_n \in \R
\]