Which integer combinations of $n$-th roots of unity are zero?

Question

Let $\omega$ be primitive $n$-th root of unity. Can we determine all tuples of integers $(c_1, c_2,\ldots,c_n) $ such that $$c_1+c_2 \omega + c_3 \omega^2+\cdots+ c_n \omega^{n-1}=0 \,?$$

It is clear to me that if $ n$ is prime, then this means $ \omega$ is a root of polynomial $$c_1 + c_2x + c_3x^2+\cdots+c_nx^{n-1} =0 \,,$$ which implies $c_1 = c_2 = c_3\cdots = c_n $ as minimal polynomial of $\omega$ in this case is $${1+x+x^2+\dotsb+x^{n-1}}\,.$$ But if $ n$ is not prime and $\phi(n) $ divides $(n-1) $ then other solutions are also possible. Does this become highly dependent on $ n$? Or can we still say something for general $ n$?

Answer 1

Many papers have been written on this question. I'd suggest having a look at

1. Conway & Jones, Trigonometric diophantine equations, Acta. Arith. 30 (1976) 229-240,

2. U Zannier, Vanishing sums of roots of unity, Rend. Sem. Mat. Univ. Pol. Torino 53 (1995) No. 4, 487-495,

3. Lam & Leung, On vanishing sums of roots of unity, Journal of Algebra 224 No. 1 (2000) 91-109,

4. Gary Sivek, On vanishing sums of distinct roots of unity, Integers 10 (2010) 365-368 #A31.