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I'm not familiar with Symmetric polynomial. As title, is it possible to express $$x^2 y^2 + y^2 z^2 + z^2 x^2$$ using $x+y+z$, $xy+yz+zx$ and $xyz$?

Is there a trick for such problems?

athos
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    $(xy+yz+zx)^2 -2xyz(x+y+z)$ – Exodd Jul 15 '20 at 21:02
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    I suggest you consult Keith Conrad's short, but excellent expository paper on the subject. In it he describes a method for expressing a symmetric polynomial in terms of the elementary ones. https://kconrad.math.uconn.edu/blurbs/galoistheory/symmfunction.pdf – Hilbert Jr. Jul 15 '20 at 21:03
  • for context, every symmetric polynomial can be written in terms of elementary symmetric functions – Exodd Jul 15 '20 at 21:04
  • Note that it is homogeneous of degree $4$ and must be some combination of expressions homogeneous of degree $4$.

    Now there are orderings of expressions of degree $4$ (which you can find in the literature) eg $x^4, x^3y, x^2y^2, x^2yz$ and you eliminate terms earliest in the list first.

    With a bit of work this becomes a systematic method.

     – Mark Bennet Jul 15 '20 at 21:07
  • you have an answer already; still, things to note ar that $x^2 + y^2 + z^2$ and $x^3 + y^3 + z^3$ and $x^4 + y^4 + z^4$ can be written using the elementary functions. I answered something using those recently, I'll check – Will Jagy Jul 15 '20 at 21:09
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    found it, shows some of the flavor https://math.stackexchange.com/questions/3749238/find-xnynzn-general-solution/3749331#3749331 Note that your target is half of $(x^2 + y^2 + z^2)^2 - (x^4 + y^4 + z^4),$ so there is a slow but sure way to get there – Will Jagy Jul 15 '20 at 21:13

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