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}

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!)