# Number of critical points of sum of two functions

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.

• Suppose that $f$,$g$ are also stable, meaning that different critical points have different critical values. This is a generic situation, i.e., probability 1 event. Consider the function $h_t=f+tg$. For $t$ very small $h_t$ has the same number of critical points as $f$ and for $t$ large it has the same number of critical points as $g$. Aug 28, 2020 at 10:41

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.