# [Polynomials] A Case of a Non-Negative Polynomial that cannot be written as a Sum of Squares

Let's show that the Motzkin polynomial cannot be expressed as the sum of squares. Assume for the sake of contradiction, that $M(x,y) = x^4y^2+x^2y^4 +1-3x^2y^2 = \sum_{j=1}^{p} f_j (x,y)^2$$=\sum (A_j+B_jx+C_jy+D_jx^2+E_jxy+F_jy^2+G_jx^3+H_jx^2y+I_jxy^2+J_jy^3)^2,$

where $$A_j,B_j,..., I_j, J_j$$ are all real constants. By comparing the above expression to Motzkin's polynomial, we see that some of the coefficients in $$\sum f_j (x,y)^2$$ equal zero, because Motzkin's polynomial does not contain some terms in $$\sum f_j(x,y)^2.$$

Firstly, Motzkin's polynomial does not have $$x^6, y^6$$, so we can eliminate the constants $$G_j$$ and $$J_j$$, that is, $$G_j=J_j=0$$.

So now $f_j(x,y)= A_j+ B_jx + C_jy + D_jx^2 + E_jxy + F_jy^2 + H_jx^2y + I_jxy^2.$

Additionally, Motzkin's polynomial does not have the terms $$x^4, y^4$$, so we know that $$D_j=F_j=0$$.

So now $f_j(x,y)= A_j + B_jx + C_jy + E_jxy + H_jx^2y + I_jxy^2.$

Likewise, there is no $$x^2$$ term in the Motzkin polynomial, which means that $$\sum B_j^2x^2=0$$, hence each $$B_j=0$$. Finally,there is no $$y^2$$ term, so the coefficient of $$y^2$$ of $$\sum f_j(x,y)^2$$ is zero, so $$\sum C_j^2 =0$$ and each $$C_j=0.$$

Therefore, we are left with $f_j(x,y)= A_j + E_jxy + H_jx^2y + I_jxy^2,$ where $$A_j, E_j, H_j, I_j$$ are constants.

Following this procedure, we see that the coefficient of the $$x^2y^2$$ term in the Motzkin polynomial is -3. This means that

$\sum 2F_jD_j+E_j^2+2B_jI_j+2C_jH_j = \sum (2F_jD_j+E_j^2)^2 = -3.$

But if we take a look at $$D_j$$, we see that there is no $$x^4$$ term in the Motzkin polynomial, so $$\sum D_j^2+2B_jG_j=0$$. But we found above that $$B_j=0$$, so $$\sum D_j^2=0$$, hence each $$D_j=0$$. Therefore, $\sum (2F_jD_j+E_j^2) = \sum E_j^2 =-3.$

Since the sum of all the $$E_j^2$$ terms cannot be negative, this is a contradiction.

Hence we conclude that the Motzkin polynomial, $$M(x,y) = x^4y^2+x^2y^4 +1-3x^2y^2$$, is an example of a non-negative polynomial of two variables that cannot be written as the sum of squares.

Previously, we showed that the Motzkin polynomial is non-negative using the AM-GM inequality. This polynomial caught our interest, because it is non-negative in $$\mathbb{R}^2$$, but cannot be expressed as the sum of two squares of polynomials. The Motzkin polynomial illustrates that although a non-negative polynomial in one variable can always be expressed as the sum of squares, this is not necessarily the case for a polynomial in two variables.

Note by Tasha Kim
10 months, 1 week ago

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