Motzkin Form

Let $M(x,y,z)$ be the Motzkin form:
$$M(x,y,z)=x^4y^2+x^2y^4+z^6-3x^2y^2z^2.$$

(a) Show that $M$ is a nonnegative form and find all of the zeroes of $M$.

(b) Calculate the Newton Polytope of the Motzkin form and use the exercise on Newton polytopes to show that the Motzkin form is not a sum of squares.


Solution:

(a) We first claim that $M$ is non-negative. This follows from the fact that the arithmetic mean of real numbers $a_1,\ldots,a_n$ is greater or equal to their geometric mean (which follows, for example, by using Jensen's inequality and the concave function $\ln (x)$). That is, for any real $a_1,\ldots,a_n$ one has
$$\frac{a_1+\ldots + a_n}{n}\geq \sqrt[n]{a_1 \cdots a_n}. \qquad (1)$$
So, in particular,
$$\frac{x^4y^2+x^2y^4+z^6}{3}\geq \sqrt[3]{x^4y^2\cdot x^2y^4 \cdot z^6}=x^2y^2z^2,$$ and the claim follows.

Furthermore, the inequality in Equation (1) is an equality if and only if $a_1=a_2=\ldots = a_n$. So, the zeros of $M$ occur exactly when $$x^4y^2=x^2y^4=z^6 \qquad (2)$$ From here it follows easily that the zeros of $M$ occur exactly along the line (in $\mathbb{R}^3$) given by $x=y=z$. In particular, note that if $M(x,y,z)=0$, then $z=0$ only if $x=y=0$, and if $z\neq 0$ then from Equations (2) we have $x\neq 0$ and $y\neq 0$ and thus also from Equations (2) we get $x=y=z$.

Q.E.D. spb

(b) The Newton polytope of $M$ is $$\mathcal{N}(M)=\operatorname{conv}\{(4,2,0),(2,4,0),(0,0,6),(2,2,2)\}.$$ We claim that $M$ is not a sum of squares. Seeking a contradiciton, and using the previous problem [exercise on Newton polytopes], we have that if $M=\sum q_i^2$ then $\mathcal{N}(q_i)\subseteq \frac{1}{2}\mathcal{N}(M)=\operatorname{conv}\{(2,1,0),(1,2,0),(0,0,3),(1,1,1)\}$, which is a subset of an affine plane in $\mathbb{R}^3$ (and looks like a triangle). In this simple case, we have $\operatorname{conv}\{(2,1,0),(1,2,0),(0,0,3),(1,1,1)\}\cap \mathbb{Z}^3=\{(2,1,0),(1,2,0),(0,0,3),(1,1,1)\},$ and so the only possible monomials appearing in the $q_i$ are $x^2y,xy^2,z^3,xyz$, which is a contradiction with the fact that the coefficient of $x^2y^2z^2$ in $M$ is negative.

Q.E.D. spb

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