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There are polynomials that are not sum of squares. For example Motzkin gave the example $x^4y^2+x^2y^4+z^6-3x^2y^2z^2$ in 1967.

Is there a real polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ in several indeterminates that is not a sum of squares but $f^N$ is a sum of squares for some odd integer $N>0$?

This question is interesting in the following sense. The notion of writing nonnegative polynomials $f$ as a sum of squares is to give an algebraic proof of the inequality $f\ge 0$. As per Motzkin's example, we know that this is not always possible. One way to resolve this is to follow Artin and use denominators. Another way (which I learnt from D'Angelo) is to show that $f^{N}$ is a sum of squares for some odd $N$.

This question is me wondering whether such a technique of consider the radical of sum of squares is vacuous.

EDIT: Let me explain why I accepted Bruce's answer over JC's. In mathematics, a variety of proofs for a single question lead us to have a more well-rounded understanding of the problem. JC's answer uses a computer, and -- without further trials and careful observation of the computer output -- only gives us an affirmative answer to my question. Bruce's answer, on the other hand, though less explicit, should be able to lead to further counterexamples by modeling after Bruce's binomial method.

I believe this method of choosing $f^2$ as a denominator for the inequality $f>0$ will lead to fruitful application in polynomial optimization and the moment problem.

4 added 672 characters in body

There are polynomials that are not sum of squares. For example Motzkin gave the example $x^4y^2+x^2y^4+z^6-3x^2y^2z^2$ in 1967.

Is there a real polynomial $f\in{\mathbb{R}}[x_1,\ldots,x_n]$ in several indeterminates that is not a sum of squares but $f^N$ is a sum of squares for some odd integer $N>0$?

This question is interesting in the following sense. The notion of writing nonnegative polynomials $f$ as a sum of squares is to give an algebraic proof of the inequality $f\ge 0$. As per Motzkin's example, we know that this is not always possible. One way to resolve this is to follow Artin and use denominators. Another way (which I learnt from D'Angelo) is to show that $f^{N}$ is a sum of squares for some odd $N$.

This question is me wondering whether such a technique of consider the radical of sum of squares is vacuous.

EDIT: Let me explain why I accepted Bruce's answer over JC's. In mathematics, a variety of proofs for a single question lead us to have a more well-rounded understanding of the problem. JC's answer uses a computer, and -- without further trials and careful observation of the computer output -- only gives us an affirmative answer to my question. Bruce's answer, on the other hand, though less explicit, should be able to lead to further counterexamples by modeling after Bruce's binomial method.

I believe this method of choosing $f^2$ as a denominator for the inequality $f>0$ will lead to fruitful application in polynomial optimization and the moment problem.

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# when is the power of a nonnegative polynomial a sum of squares?

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