# Non-negative polynomials and sums of squares

Let $\mathcal{P}_{n,2d}$ denote the real forms of degree $2d$ in $n$ variables and $\Sigma_{n,2d}$ the forms that can be written as sums of squares. The Motzkin-polynomial shows that $\mathcal{P}_{3,6}\neq\Sigma_{3,6}$. Starting with this polynomial, how does one get counter examples for higher degrees or higher number of variables?

So first for higher number of variables, you can take again the Motzkin polynomial $M$: Assume that $M$ is a sum of squares of polynomials in $x_1, \ldots, x_n$ with $n>3$. Then by setting $x_4=\ldots=x_n=0$ we get a sum of squares representation of $M$ with polynomials in $x_1,x_2,x_3$ which does not exist.
For higher degree consider $P=x_1^{2d} \cdot M$. If $P=g_1^2+\ldots+g_r^2$ for some polynomials $g_i$, then each $g_i$ must vanish on the hyperplane $x_1=0$. Thus every $g_i$ is divisible by $x_1$ and by dividing both sides by $x_1^2$ you get a sum of squares representation of $x_1^{2d-2}\cdot M$. You iterate this until you have a sum of squares representation of $M$ which is a contradiction.