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

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 $$n$$?

• Obviously, we're going to answer it. But what are your thoughts regarding the question. Jul 22, 2020 at 6:19
• Hint: This isn't as much about roots of unity as it is about vectors in the plane. Jul 22, 2020 at 6:20
• @Arthur is correct. Notice that $n^{th}$ roots of unity lie on vertices of $n$-sided regular polygon with one side along real axis. Jul 22, 2020 at 6:26
• Answer depends on $n$. Easy case is when $n$ is prime. Can you do that case, Aditya? By the way, perhaps you want to specify that $\omega$ is a primitive $n$th root of unity? Jul 22, 2020 at 6:28
• @Arthur Is it really? It’s basically asking for any linear relations within the $n$-th roots of unity and linear algebra alone doesn’t seem to get you far without knowing a thing or two about the roots first, does it? Jul 22, 2020 at 6:31

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.

• This should be written in comments. Jul 24, 2020 at 14:54
• If you have a better answer, @SarGe I invite you to post it. Jul 25, 2020 at 3:16
• Henry B Mann , ON LINEAR RELATIONS BETWEEN ROOTS OF UNITY ,1965 This paper seems also related to the question. Jul 26, 2020 at 17:07
• @Alex, yes, it's probably in the references of the other papers so I figured anyone who followed the links I gave would come across the Mann paper. Do you have a link to it online? Jul 26, 2020 at 23:10
• Jul 26, 2020 at 23:48