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I've encountered the following problem during my research, any help would be highly appreciated.

Let a following multivariate equation be given:

$F(x,P,I) = 0$,

where $x$ is a variable and $P,I$ are disjoint sets of other variables. Say, for example, that this equation is polynomial for all the variables.

I want to find such conditions $G_k (P,I) = 0, k=1,...,n$ with which $x^{\ast} = H(P)$ would be a solution to $F(x^{\ast},P,I)$.

In other words, I want to find a solution $x^{\ast}$ that does not depend on parameters $I$. Naturally, in general, this will only be possible if there are some conditions $G_k (P,I) = 0$ on $P$ and $I$, so I'd like to find such conditions also.

Has any work been done on problems like this? I'd appreciate any point in the right direction.

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  • $\begingroup$ For polynomials, this is called elimination theory. $\endgroup$
    – Ben McKay
    Commented Oct 9, 2022 at 13:03
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    $\begingroup$ You want $f=F(H(P), P, I)$ to be zero for all values of $I$, which implies $f$, $df/dI$, $d^2\!f/dI^2$ etc are all identically zero. Requiring that all those derivatives are 0 at some convenient arguments is usually sufficient (and always sufficient if $F$ and $H$ are both polynomial). This approach can also allow you to iteratively determine the coefficients of $H$. But if you want more details, it’d be helpful to give a sample problem like this which you know how to solve, and another sample problem which you don’t know how to solve. $\endgroup$
    – user44143
    Commented Oct 9, 2022 at 16:34

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