# How to solve equations over polynomial rings

sorry if my question is very basic but I don't know what to even search to look it up and the only "obvious" places I thought of had nothing.

Some background, for whatever context it might provide. I'm working with a certain ring of functions $R=\langle P_i\rangle$, where $P_i=ax^i+y^i+z^i+(-1)^i (ax+y+z)^i\in\mathbb{C}(a)[x,y,z]$ (i.e. I normally think of $a$ as a coefficient of $x$, but here I'm actually keeping it as a variable, hoping to generalize a calculation I'll explain below). By a theorem of Hilbert's, I know that $R$ is finitely generated. More importantly, if it is free over $\mathbb{C}[P_2, P_3, P_4]$, it is Cohen-Macaulay and that's what I want to know ultimately. If I compute the Hilbert series and all coefficients are positive, then this tells me where I should pick generators from. Sorry if this is more information than is relevant.

OK, well in particular, say I think the generators should be $1, P_4, P_5$, meaning that polynomials like $P_{10}$ should have a (unique) expression of the form $f_1(P_2, P_3, P_4)+f_2(P_2, P_3, P_4)P_4+f_3(P_2, P_3, P_4)P_5=P_{10}$, where the $f_i$ are polynomials. I don't care to show the uniqueness, but given all the $P_i$, is there any systematic way to have Mathematica solve for the polynomials $f_i$? The bounds on degrees give some pretty tight bounds on the degrees of the $f_i$.

Any help would be much appreciated!

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Are you sure you want P4 both defining the ring and as a generator? –  Daniel Lichtblau Jul 17 '14 at 16:02