A sketch of the proof of the following statement is in Barvinok's "A Course in Convexity".
Fixing positive integers $k$ and $n$, define $H_{2k,n}$ to be the real vector space of all homogeneous polynomials $p(x)$ of degree $2k$ in $n$ real variables. Then, if $p\in H_{2k,n}$ is a positive polynomial, there exists a positive integer $s$ and vectors $c_1,\cdots,c_m\in\mathbb{R}^n$ such that $$\|x\|^{2s-2k}p(x)=\sum_{i=1}^m\langle c_i,x\rangle^{2s} ~~ \mbox{for all} ~~ x\in\mathbb{R}^n.$$
Here, given $c=(\lambda_1,\cdots,\lambda_n)\in\mathbb{R}^n$, the notation $\langle c,x\rangle$ is used to denote the linear polynomial $\sum_i\lambda_ix_i\in H_{1,n}$.
Note that by 'homogeneous polynomial', we are simply referring to a polynomial in which the (nonzero) terms all have the same degree; and by 'positive polynomial', we are simply referring to a polynomial $p$ for which $p(x)>0$ for all $x\in\mathbb{R}$.
My question is: why does this claim fail if "positive polynomial" is replaced with "non-negative" polynomial? I've tried to parse the proof to see where strict positivity is required, but the proof is dense. It seems like there should be a good counterexample. This is a homework question, so I appreciate hints.
Edit: I am able to show that the Motzkin polynomial ($p(x,y)=x^4y^2+x^2y^4-3x^2y^2+1$) is a counterexample, but I am still wondering what the intuition is on why the claim fails for non-negative polynomials.
Edit Actually the above is not a counterexample as that is not a homogeneous polynomial.