Let $F(\vec{t}) = R_1(\vec{t}) = 0$ and $G(\vec{s}) = R_2(\vec{s}) = 0$ be two functions where $F,G: \mathbb{R}^n \rightarrow \mathbb{R}^m$ and $R_1,R_2$ are rational.
Overall, my goal is to find those parameters $\vec{t} = \vec{s}$ where both functions are satisfied. However, I'm not convinced that this is possible in general. Instead, I'd like to explore the sets of parameters for which both functions are simultaneously satisfied. For instance, is the set empty? How large is it? Can I bound it?
Can anyone provide me with some guidance on what to start looking for to begin understanding questions like these?