# Intersection of two hypersurfaces via... Bezout's theorem?

Let $$M_1,M_2 \subset \mathbf{S}^n$$ be two smoothly embedded, connected hypersurfaces of the round sphere, which are realized as the zero sets of two homogeneous polynomials $$P_1,P_2$$ in $$\mathbf{R}^{n+1}$$: $$$$M_i = \{ P_i = 0 \} \cap \mathbf{S}^n.$$$$

Is there an upper bound for the number of connected components of $$M_1 \setminus M_2$$ in terms of $$\operatorname{deg} P_1$$ and $$\operatorname{deg} P_2$$?

I apologize for the elementary question; my impression is that it might follow from Bezout's theorem, perhaps combined with Mayer–Vietoris, but I'm not an algebraic geometer by trade, so I can't say for certain. (I'd be happy to look up references myself if you have a suggestion!)

• Are you looking for real intersection points or complex ones? Commented Feb 15, 2023 at 8:45
• Real - I guess it's basically a differential geometry question, just that the hypersurfaces considered happen to be algebraic. (Is what I said about $M_1 \cap M_2 \neq \emptyset$ false in this case, because the field isn't algebraically closed?) Commented Feb 15, 2023 at 8:51
• Think about two very thin quadratic cones, so their zeroes intersect the sphere in tiny circles, which don't have to intersect; they can be any circles, appearing in antipodally symmetric pairs. Commented Feb 15, 2023 at 8:53
• OK, you're right - I'll update the question. I'm honestly mostly interested in an upper bound for the number connected components, so perhaps this is still OK as a question? Commented Feb 15, 2023 at 8:58
• Such upper bounds exist, just by "quasi-compactness." I guess you are looking for an effective upper bound. Commented Feb 15, 2023 at 16:13

In Proposition 4.13 of Coste’s introduction to semi-algebraic geometry, a bound of $$d(2d-1)^{s+k-1}$$ is given for the number of connected components of a system of $$s$$ real polynomial equations and inequations of degree at most $$d\ge 2$$ in $$k$$ variables. In your case, $$k=n+1$$ and $$s=3$$.
• Cool, thanks! Yowza, those bounds are brutal! I'll leave the question open for now, perhaps someone sees a shortcut that uses the fact that $M_i = P_i^{-1}(0) \cap \mathbf{S}^n$ are both connected. I hope you won't take offence! (In the meantime, I'll check if the library has a copy of the book; thanks for the suggestion.) Commented Feb 15, 2023 at 19:15
• OK, it looks like nobody's biting - thanks again for your answer! I'm still a bit taken aback that the bounds would so bad; my expectation was a bound by $\operatorname{deg} P_1 \operatorname{deg} P_2$, but perhaps I was a bit naive... Commented Feb 16, 2023 at 23:18