Vanishing sums of roots of unity

A root of unity is a complex number z such that z^n=1 for some positive integer n. This means that z=e^{2\pi i k/n} for some integer k; if k is relatively prime to n, we say that z is a primitive n-th root of unity, meaning that z is not a m-th root of unity for any 1\leq m<n.

Here’s a question: when can we have

(1) \ \ z_1+\dots +z_\ell=0

if z_1,\dots,z_\ell are roots of unity?

This is a little bit vague, in that I did not say what kind of tentative characterization we are looking for. If you were inclined to play devil’s advocate, you could say that equation (1) provides a good enough description. There are, however, less obvious answers that have come in handy in various parts of my research, so let’s look at some of them.

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