I ran into the following "simple" question and I am wondering whether there are any references, which might help me. I am coming from statistics, so I am not so aware which branch of math could have already dealt with this question.


Let $S\subset\mathbb{R}^D$ be compact and assume that $f,g: S\rightarrow \mathbb{R}$ are $C^2$ Morse functions (so basically requiring finitely many critical values). Further assume that $f+g$ has only finitely many isolated critical points.
Denote with $C\_f$ and $C\_g$ the number of critical points of $f$ and $g$.


Is it possible to find a bound on $C\_{f+g}$ depending only on $C\_f, C\_g$ and $D$ and maybe $S$?


For polynomials this is easy and simply Bezout's Theorem, yet I need something like it for more general functions.



Number of critical points of sum of two functions
Is the following example helpful? This is inspired by the fact that no bound on the "degree" of the two functions $f$ and $g$ is assumed. As far as I understand Bezout's theorem, this would make a bound difficult even for polynomial functions.


Here let $A,B > 1$ be large and $0 < \epsilon < 1$ be small. Let $D = 1$, and $S = [0,1]$ be the closed unit interval. Define $f: x \mapsto Ax + \epsilon \cos(Bx)$ and $g: x \mapsto Ax$. Then $f'(x) = A - B\epsilon \sin(Bx)$ and $g'(x) = A$, so that neither function has critical points in $[0,1]$ provided just that $A > B \epsilon$. However their difference $f-g$ has derivative $(f-g)'(x) = - \epsilon \sin(Bx)$. By taking suitably large $A,B$ this function has arbitrarily many critical points in $[0,1]$.


Perhaps it is worth pointing that really only $B$ needs to be large in the argument above. In particular, by first picking $B$ large and then a small $\epsilon > 0$ in terms of it, one is free to pick a small $A$ as well. Therefore the functions $f,g$ can be taken to have arbitrarily small first and second derivatives.


