  In this note we aim to give a new, elementary proof of a statement that was
first proved by Timofte. It says that a symmetric real polynomial $F$ of degree
$d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only
if it is so on the subset of points with at most $\max\{\lfloor d/2\rfloor,2\}$
distinct components. We deduce Timofte's original statement as a corollary of a
slightly more general statement on symmetric optimization problems. The idea we
are using to prove this statement is to relate it to a linear optimization
problem in the orbit space. The fact that for the case of the symmetric group
$S_n$ this can be viewed as a question on normalized univariate real
polynomials with only real roots allows us to conclude the theorems in a very
elementary way. We hope that the methods presented here will make it possible
to derive similar statements also in the case of other groups.
