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Show that the polynomial $x^4y^2+y^4z^2+z^4x^2-3x^2y^2z^2$ cannot be written as the sum of squares of polynomials over $\mathbb{R}$ in $x, y, z$.

I could not make any progress/significant observation except for showing that the polynomial is always non-negative.

Subham Jaiswal
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1 Answers1

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The proof is similar to the case of the Motzkin polynomial

$$ x^4y^2 + x^2y^4 - 3x^2y^2 + 1,$$

which is non-negative over $\mathbb{R}^2$ and yet cannot be expressed as a sum of squares of real polynomials in $x$ and $y$. For references and proofs (e.g. by M. Marshall) see here.

Dietrich Burde
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  • Sir, can you give an elementary proof of my problem? I am in high school and the problem is from E. J. Barbeau polynomials. – Subham Jaiswal Aug 05 '15 at 14:01
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    There is a high school proof available, see here, for the polynomial $M(x,y,z)=x^4y^2+x^2y^4+z^6-3x^2y^2z^2$. Just do the same for your polynomial. – Dietrich Burde Aug 05 '15 at 14:05