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I want to simplify this homogeneous polynomials: $$x^3 + y^3 + z^3 + x^2 y + x y^2 + x^2 z + x z^2 + z^2 y + z y^2 + x y z$$ to a simpler form. For example, the homogeneous polynomials of degree 2 can be simplified as follows: $$x^2 + y^2 + z^2 + x y + x z + y z = \frac{1}{2}((x + y)^2 + (x + z)^2 + (y + z)^2)$$. So I wonder whether degree of 3 could also be simplified. I know this kind of "simplification" is not very well-defined and very dependent on experience. I have tried mathematica but in vain. Is there any simpler form?

Michael Rozenberg
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yzxhd
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  • See also this question. I saw some closer ones but cannot find them now. I obtain $(y + z + x)x^2 + (y^2 + z^2)(y + z) + (y^2 + yz + z^2)x$, but it doesn't seem to be better. – Dietrich Burde Mar 19 '20 at 20:39
  • I don't believe there is much hope for this, at least not in a similar form. When we calculate $(x+z)^3 +(y+z)^3 + (x+y)^3$, we get differing coefficients, so dividing by a single number can't do the trick. We would need to adjust by subtracting off something, and then dividing. Maybe exploring this further is worthwhile though. – Chaotic Good Mar 19 '20 at 20:42

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Maybe $$(x^2+y^2+z^2)(x+y+z)+xyz$$ or $$\frac{x^5y+y^5z+z^5x-x^5z-y^5x-z^5y}{(x-y)(x-z)(y-z)},$$ where $\prod\limits_{cyc}(x-y)\neq0?$

Michael Rozenberg
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