Solving complex exponential polynomials

[beechner224]

Are there any general methods to solve the following complex exponential polynomial without relying on numerical methods? I want to find all possible solutions, not just a single solution.

e^(j*m*$$\theta1$$) + e^(j*m*$$\theta2$$)+e^(j*m*$$\theta3$$) + e^(j*m*$$\theta4$$) + e^(j*m*$$\theta5$$) = 0

where

$$\theta1$$<$$\theta2$$<$$\theta3$$<$$\theta4$$<$$\theta5$$

and

m is a integer

 

[Gerenuk]

At first a question:
Why do you include m? You could absorb it into the thetas?
The ordering of the thetas probably doesn't play a role either.

With that in mind you could probably solve these equations this way:
The sum of three terms should be smaller than 2 in magnitude. Then it is always possible to find the final two exponentials.

 

[Count Iblis]

http://en.wikipedia.org/wiki/Pentagon