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 $\mathbf{R}^{n+1}$:
\begin{equation}
M\_i = \{ P\_i = 0 \} \cap \mathbf{S}^n.
\end{equation}



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> 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 $\operatorname{deg} P\_2$?
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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!)



Intersection of two hypersurfaces via... Bezout's theorem?
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 $s=3$.


