Consider a finite Fourier sum of the form $$f(\theta) = \sum_{i=1}^n r_i \cos(m_i \theta) \,,$$ where $n \geq 1$ is an integer, the $r_i$ are positive real numbers and the $m_i$ are integers. Is there a formula which bounds by above the amount of distinct minima of $f$ in terms of the $r_i$ and $m_i$?
More generally, is there a formula which bounds by above the amount of distinct minima of the function $$F(\boldsymbol{\theta}) = \sum_{i=1}^n r_i \cos(m_{ij} \theta^j) \,,$$ where $\boldsymbol{\theta} = (\theta^1, \theta^2, \cdots, \theta^l)$ is an $l$-vector, $l \geq 1$, and the repeated index $j$ should be summed over from $j = 1$ to $l$? Likewise the $m_{ij}$ are integers and the $r_i$ are positive real numbers.
What I have so far, which is not completely satisfactory, and applies only to the $l = 1$ case: the function $f'$ is a trigonometric polynomial of degree $N = \max m_i$. Such functions have at most $2N$ distinct zeroes, so this number also bounds the amount of distinct minima of $f$. (Powell p.150, Approximation Theory and Methods (1981), Cambridge University Press)