<s>[INST] Writing $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\frac1x+\frac1y+\frac1z$ Can we write $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\dfrac1x+\dfrac1y+\dfrac1z$?
What about $x^3+y^3+z^3$?
 [/INST] Yes, by the Fundamental Theorem of Symmetric Polynomials since $x^2+y^2+z^2$ is symmetric and $xy+yz+xz=xyz*(1/x+1/y+1/z).$
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