I seem to recall that a general *homogenous* real polynomial $P$ of even degree in $n$ variables, $n\geq 3,$
cannot always be expressed as $P(x_1,\dotsc,x_n)=\sum_j a_j Q_j^2(x_1,\dotsc,x_n)$ where $a_j \in \mathbb{R},$ and the $Q_j^2$ are *homogenous of the same degree* as $P.$
(Please, correct me if I am wrong).

Now, what if we know that $P$ is symmetric? Is it still true that a general polynomial cannot be expressed as above? And if it can, what if we require that the $Q_j$ themselves are symmetric?

**Question:** If $P$ is symmetric and homogenous of even degree,
can it be expressed as a sum/difference of squares of homogenous and symmetric polynomials?

I know that if $P$ is symmetric, and $P(x_1+t,x_2+t,\dotsc,x_n+t)=P(x_1,\dotsc,x_n),$ then there is always such a representation as the one above, but can one lose the translation-invariant condition and still have sum-and-difference of squares representation?