Can every symmetric polynomial of degree $r$ in $d$ variables that has no constant term be written as a sum of the $r$th powers of linear polynomials in $d$ variables and a homogeneous polynomial of degree $r$ each of whose terms involves at most $d−1$ variables?


The linear polynomials are truly linear functions: e.g. $p(x) = w^T x,$ so have no constant terms. And the field is $\mathbb{R}.$


As an example, this is true for degree two polynomials, because you can show that given an arbitrary $A$, there exists a nonnegative matrix $B$ and a diagonal matrix $D$ such that the quadratic forms corresponding to $A$ and $B+D$ are identical.



Decomposition of symmetric homogeneous polynomials
This is true if $r$ is odd or $r<2d$, and false otherwise. I assume, when you say a polynomial is symmetric you mean in fact it is homogeneous.


**1.** Let $r=2d$ and take $P(x)=-x\_1^2x\_2^2\dots x\_d^2$. The term $x\_1^2\dots x\_d^2$ appears in every $r$th power of a linear form with a nonnegative coefficient, so their sum plus monomials in $d-1$ variables cannot get a negative coefficient at $x\_1^2\dots x\_d^2$. A similar reasoning works for all even $r\geq 2d$.


**2.** Now assume that $r$ is odd or $r<2d$. In the former case we will even show that each appropriate polynomial is even a sum of $r$th powers.


Firstly, notice that 
$$
 (-1)^rr!y\_1y\_2\dots y\_r=\sum\_{I\subset\{1,\dots,r\}}(-1)^{|I|}
 \left(\sum\_{i\in I}y\_i\right)^r.
$$
So the monomial $y\_1\dots y\_r$ is a linear combination of $r$th powers. By substitution, the same holds for every $r$th degree monomial in $x\_1,\dots,x\_d$. So the linear hull of all $r$th powers is the whole space of all homogeneous polynomials of degree $d$.


Let $S^{d-1}$ be a unit sphere in $\mathbb R^d$, and let $\mu$ be the usual measure on it. If $r$ is odd, then
$$
 \int\_{S^{d-1}}(a^Tx)^r\,d\mu(a)=0,
$$
so $0$ lies in the convex hull of all the polynomials $(a^Tx)^r$. By Caratheodory's theorem, $0$ is a finite convex combination of some of them, and (due to symmetry of the sphere) each $r$th power is involved in some such combination. This yields that $0$ lies in the (relative) interior of the cone generated by $r$th powers, so this cone is the whole space. This is exactly what we claimed.


(Some time ago I was told that this Caratheodory trick appeared in a paper by Milman and someone else; sorry for not presenting the exact reference.)


If $r$ is even but $r<2d$, then the same integral is a nonzero polynomial $Q$, but it contains only monomials of even degree in each variable; so it is a linear combination of monomials in at most $d-1$ variables (and it is invariant under sphere rotations). Then the same arguments can be applied to the factor space by $\langle Q\rangle$. We will represent each polynomial as a sum of $r$th powers plus a multiple of $Q$, which is again what we need.


