Non-negative polynomials and sums of squares

Question

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?

Answer 1

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.