Let $f : [0,2\pi]^d \to \mathbb R$ be multivariate trignometric polynomial of the form

$$ f(x_1, \cdots, x_d) = \sum_{i=1}^n a_i \prod_{j=1}^d f^{i}_j(x_j), \quad a_i \in \mathbb R $$

Where each $f^i_j$ is a trigonometric function, such as sine or cosine. I am interested in studying the (number of) connected components of the level sets of multivariate trigonometric polynomials.

Do you have any advice on relevant literature and available techniques? Is there something analogous to real algebraic geometry (smooth manifold theory in case the level sets are smooth manifolds) for multivariate trigonometric polynomials that could be used here? Any techniques available to upper bound the rank of $H^0$ (zeroth cohomology) of level sets defined by such multivariate polynomials.