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?



Homogenous polynomials as sum or differences of squares and symmetric polynomials
My college provided me with a simple example:
$$x^2 + xy + y^2$$ cannot be expressed as a sum/difference of symmetric, homogenous polynomials of degree 1, for obvious reasons.


(Each homogenous, symmetric polynomial of degree 1 in two variables are of the form a(x+y).
Thus, all possible $Q\_j$ are of the form $a'(x+y)^2$ which, of course, cannot give the polynomial above.)


