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Suppose a polynomial $P(x,y)$ is given and it is to be written in terms of two base polynomials $R(x,y)$ and $S(x,y)$. How does Mathematica do this?

The assumption is that $P$ will turn out to be a polynomial in $R$ and $S$. For example $P=x^6+3x^4y^2+3x^2y^4+y^6+5x^2y^4$ and $R=x^2+y^2$ and $S=xy^2$. The program is to produce $P=R^3+5S^2$.

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    $\begingroup$ Also: Solve[{p == x^6 + 3 x^4 y^2 + 3 x^2 y^4 + y^6 + 5 x^2 y^4, r == x^2 + y^2, s == x y^2}, {p}, {x, y}] -- GroebnerBasis[{x^6 + 3 x^4 y^2 + 3 x^2 y^4 + y^6 + 5 x^2 y^4, r - (x^2 + y^2), s - x y^2}, {r, s}, {x, y}] -- Eliminate[{p == x^6 + 3 x^4 y^2 + 3 x^2 y^4 + y^6 + 5 x^2 y^4, r == x^2 + y^2, s == x y^2}, {x, y}] $\endgroup$ – Michael E2 May 19 '17 at 21:32