Is there a general approach to solve a $f(x) \in R[x]$ which satisfy an equation like $a_0(x)+a_1(x)f(x)+a_2(x)f^2(x)+... = 0$ where $a_i(x)\in R[x]$, $R$ is a ring.
Further, is there a general approach to solve a $f(x_1,...x_n) \in R[x_1,...x_n]$ which satisfy an equation like $a_0(x_1,...x_n)+a_1(x_1,...x_n)f(x_1,...x_n)+a_2(x_1,...x_n)f^2(x_1,...x_n)+... = 0$ where $a_i(x_1,...x_n)\in R[x_1,...x_n]$.
Finally, is there a general approach to solve a vector $[f_1(x_1,...x_n),...,f_m(x_1,...x_n)] \in (R[x_1,...x_n])^m$ which satisfy an equation like $\sum_a a(x_1,...x_n)f_1^{a_1}(x_1,...x_n)...f_m^{a_m}(x_1,...x_n) = 0$ where $a(x_1,...x_n)\in R[x_1,...x_n]$.