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Invariant theory deals with an algebraic, geometric or analytic structure $X$, submitted to the action of an (algebraic) group $G$. It studies $G$-invariant elements of $X$ as well as the set of $G$-orbits.

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Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

I do not know if it is related to your problems, but there are two papers of R. Dvornicich and U. Zannier studying the field of fraction generated by subsets of the P_i in the case p,q and r are equal …
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