Let ${\boldsymbol \theta}=(\theta_1,\theta_2,\ldots,\theta_n) \in{\mathbb T}^n$ and $P:{\mathbb T}^n\rightarrow {\mathbb R}$ be a function defined on $n$-torus as
$$
P({\boldsymbol \theta}) = \sum_{i<j}(1+\cos(\theta_i-\theta_j))^2.
$$
What are **local** maximum points of $P$?

One can simply show that the global maximum is ${\boldsymbol \theta} = (\theta,\theta, \ldots,\theta)$ for all $\theta \in {\mathbb S}^1$, but the question is regarding the *local* maximum points of it.

explicitlylocate all the local maxima? There are vaguely similar innocuous functions on higher tori where locating the minima could solve the notorious circulant Hadamard conjecture. I vote to re-open $\endgroup$ – Yemon Choi Aug 18 '13 at 19:38