This is a follow-up question to this question. In that question, we learned that if $T(y) = \mathrm{Res}_{x}(f(x), y - g(x))$ , then $T(g(z))$ is divisible by $f(z)$. More specifically, we can say that if $\alpha \in Z(f) \iff g(\alpha) \in Z(T)$, where $Z(f) = \{x : f(x) = 0\}$.
Now, my question is:
If $T(y) = \mathrm{Res}_{x}(f(x), y \cdot h(x) - g(x))$, where $f(x), g(x), h(x) \in R[x]$ and $Z(f) \cap Z(g) = \phi$ and $Z(f) \cap Z(h) = \phi$, where $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, where $p$ is any prime and $k \geq 2$. Now, Can we get some information about the zeros of $T$?
We know that, there exists two polynomials $a(x,y), b(x,y) \in R[x,y]$ such that $T(y) = a(x,y)f(x) + b(x,y)(y \cdot h(x) - g(x))$. Is there any way to find the root of $T$? Also if we consider $\alpha \in Z(f)$, then we can write that $T(y) = b(\alpha,y)(y \cdot h(\alpha) - g(\alpha))$.
Any kind of help is welcome.
