Is there a result like Harnack's curve theorem for real trig polynomials which gives bounds for the number of connected components the zero set can have in terms of the degree?
More specifically, let $C$ be the curve in $[0,1]^2$ given by the level set $f(x,y) = 0$ where
$$f(x,y) = a_{0,0} + \sum_{k=1}^n \sum_{l=1}^n a_{k,l} \cos(2 \pi (kx + ly)) + b_{k,l} \sin(2 \pi (kx + ly)),\text{ with }a_{k,l},b_{k,l}\in \mathbb{R},$$
and let $c$ be the number of connected components of $C$. Then can we establish a bound of the form $c = O(n^2)$ or something of the like?