Is there a bound on the number of connected components of a zero set of an integrable function?

Question

If $f$ is a real-analytic function on $[0,1]^n$, and $f$ has finite differential transcendence degree, is there some way to bound the number of connected components of its zero set or the set where it is positive in terms of $n$, the differential transcendence degree of $f$, the degree of polynomials in a characteristic set for $f$, and the sizes of the coefficients in the polynomials in the characteristic set? I am thinking of the example $f(x)=\cos(\alpha 𝑥)$ for some real $\alpha$, where one always has $𝑓″=−\alpha^2𝑓$ but, as $\alpha$ becomes large, the number of connected components goes to infinity in a polynomial way in terms of $\alpha$.

There is a result like this for Pfaffian functions, where the number of connected components can be bounded by the Pfaffian format and degree of $f$ -- proved here by Zell -- but I do not know of any result that works in this more general setting.

Answer 1

In your example, you can eliminate $\alpha$ by one more differentiation, and obtain that $f=\cos\alpha x$ satisfies $$f'''f-f''f'=0,$$ so the number of components of the zero set cannot be estimated in terms of quantities that you listed.