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Does there exist a quaternary quartic $f$ (a form in $\mathbb{R}[x_1,x_2,x_3,x_4]$ of degree $4$), which is positive semi-definite ($f \geq 0$ on $\mathbb{R}^4$) but not a sum of squares, such that one variable in $f(x_1,x_2,x_3,x_4)$ has only even exponents?

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Yes, there is such an example in [Choi, Positive semidefinite biquadratic forms, §4]. It is proven there that the polynomial $$x^4+y^4-2(x^2+y^2)zw+z^2w^2+2(x^2z^2+y^2w^2)$$ is positive semidefinite, but not a sum of squares.

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