I am trying to tackle the following family of systems of (d-1) (complex) equations:
$$\sum_{j=0}^{d-1} \exp\Bigl(i (\theta_{j+r} - \theta_{j})\Bigr) = \frac{d}{\sqrt{2}} $$
where:
- $2 \le d \in \mathbb N$
- $\theta_i \in [0,2 \pi), \; \forall \, i=0,\dots ,d-1$ are the unknown phases
- Each $r = 1, \dots , (d-1)$ corresponds to a different equation
- Subscript addition is understood $(j+r) \mod d$
I am interested first of all knowing whether a solution exists for all integers $d \ge 2$ and also finding explicit solutions.
Example: For $d=3$ the 2 equations read
\begin{cases} e^{i(\theta_2-\theta_1)} + e^{i(\theta_1-\theta_0)} +e^{i(\theta_0-\theta_2)} = \frac{3}{\sqrt{2}} \\ e^{i(\theta_2-\theta_0)} + e^{i(\theta_1-\theta_1)} +e^{i(\theta_0-\theta_2)} = \frac{3}{\sqrt{2}} \\ \end{cases}
for which a solution can be easilly obtained.
Clearly, any cyclic permutation of a sulution $(\theta_0,\dots,\theta_{d-1})$ produces another solution, as well as multiplication by a phase $e^{i \phi}$
My Progress: I was able to show (by trying a specific ansatz) that there exist solutions up to $d=13$.
Any help is appreciated!