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][1] by Zell -- but I do not know of any result that works in this more general setting.

  [1]: https://doi.org/10.1016/S0022-4049(99)00017-1 "Zell, Thierry, Betti numbers of semi-Pfaffian sets, J. Pure Appl. Algebra 139, No. 1-3, 323-338 (1999). zbMATH review at https://zbmath.org/0968.14032"