TITLE: When do complex number vectors add up to $0$?
QUESTION [0 upvotes]: I came across a few problems in which three vectors in a triangle(which were also solutions to a cubic equation) or $6$ complex numbers - (heptagon - $7$ roots of unity) added up to $0$. When does this happen? Is it only when the numbers are solutions to an equation?
When do complex number vectors add up to $0$?

REPLY [0 votes]: Every finite set of complex numbers $\{z_1,z_2,\ldots,z_n\}$ is the set of roots of an equation:
$$ (z - z_1)(z - z_2)\cdots(z - z_n) = 0. $$
To get a set of $n$ complex numbers whose sum is $0,$
begin with any set of $n-1$ complex numbers
$\{z_1,z_2,\ldots,z_{n-1}\}.$
Then set
$$ z_n = -(z_1+z_2+\cdots+z_{n-1}). $$
There are certain highly symmetric sets of complex numbers that form regular polygons around $0$ in the complex plane, and those have interesting equations.
But you don't need any kind of symmetry to have the numbers add up to $0.$
They just need to add that way.
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