Sums of Squares over Rationals
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$$\small
\begin{array}{rl}
F=&x^8+2x^6z^2-2x^5z^3-4x^5zy^2+3x^4z^4+9x^4z^2y^2+x^4y^4-2x^3z^5-16x^3z^3y^2\\\\
&-6x^3zy^4+3x^2z^6+x^2z^4y^2+20x^2z^2y^4+x^2y^6-2xz^7-4xz^5y^2-8xzy^6+z^8\\\\
&+z^6y^2+z^4y^4+y^8\\\\
\sim&(x^4+0.4271x^2z^2-1.4271xy^2z+xz^3+0.3568y^4-0.5y^2z^2-0.7454z^4)^2 +\\\\
&(-1.0704x^3z+0.5352x^2y^2+1.8684x^2z^2-3.7368xy^2z-0.3819xz^3+0.9342y^4+0.1910y^2z^2+0.6666z^4)^2
\end{array}
$$