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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?

Michael Stachowsky
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  • This sounds like a question in algebraic geometry. Are $F$ and $G$ also rational functions? I also note that the title of the question doesn't match the question. – Greg Martin Nov 08 '16 at 16:13
  • I wasn't sure if it fit with algebraic geometry or not. A good example here would be the parameterization of sine and cosine. The two functions are not polynomial and can be parameterized using rational functions. I'm not exactly sure what that gives me (ie: whether there is a "nice" way to go from these rational functions to polynomials) – Michael Stachowsky Nov 08 '16 at 16:23

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