[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)=x4y2+x2y4+13x2y2=j=1pfj(x,y)2M(x,y) = x^4y^2+x^2y^4 +1-3x^2y^2 = \sum_{j=1}^{p} f_j (x,y)^2 =(Aj+Bjx+Cjy+Djx2+Ejxy+Fjy2+Gjx3+Hjx2y+Ijxy2+Jjy3)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 Aj,Bj,...,Ij,JjA_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 fj(x,y)2\sum f_j (x,y)^2 equal zero, because Motzkin's polynomial does not contain some terms in fj(x,y)2.\sum f_j(x,y)^2.

Firstly, Motzkin's polynomial does not have x6,y6x^6, y^6, so we can eliminate the constants GjG_j and JjJ_j, that is, Gj=Jj=0G_j=J_j=0.

So now fj(x,y)=Aj+Bjx+Cjy+Djx2+Ejxy+Fjy2+Hjx2y+Ijxy2.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 x4,y4x^4, y^4, so we know that Dj=Fj=0D_j=F_j=0.

So now fj(x,y)=Aj+Bjx+Cjy+Ejxy+Hjx2y+Ijxy2.f_j(x,y)= A_j + B_jx + C_jy + E_jxy + H_jx^2y + I_jxy^2.

Likewise, there is no x2x^2 term in the Motzkin polynomial, which means that Bj2x2=0\sum B_j^2x^2=0, hence each Bj=0B_j=0. Finally,there is no y2y^2 term, so the coefficient of y2y^2 of fj(x,y)2\sum f_j(x,y)^2 is zero, so Cj2=0\sum C_j^2 =0 and each Cj=0. C_j=0.

Therefore, we are left with fj(x,y)=Aj+Ejxy+Hjx2y+Ijxy2,f_j(x,y)= A_j + E_jxy + H_jx^2y + I_jxy^2, where Aj,Ej,Hj,IjA_j, E_j, H_j, I_j are constants.

Following this procedure, we see that the coefficient of the x2y2x^2y^2 term in the Motzkin polynomial is -3. This means that

2FjDj+Ej2+2BjIj+2CjHj=(2FjDj+Ej2)2=3.\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 DjD_j, we see that there is no x4x^4 term in the Motzkin polynomial, so Dj2+2BjGj=0\sum D_j^2+2B_jG_j=0. But we found above that Bj=0B_j=0, so Dj2=0\sum D_j^2=0, hence each Dj=0D_j=0. Therefore, (2FjDj+Ej2)=Ej2=3.\sum (2F_jD_j+E_j^2) = \sum E_j^2 =-3.

Since the sum of all the Ej2E_j^2 terms cannot be negative, this is a contradiction.

Hence we conclude that the Motzkin polynomial, M(x,y)=x4y2+x2y4+13x2y2M(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 R2\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 Bright Glow
3 years, 1 month ago

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