What symmetries arise among $n$ complex numbers on the unit circle if the sum of their squares is zero?

Question

Suppose I had a list of $N$ complex numbers $\{z_i\}$ arrayed on the unit circle. If it is required that this set obeys $$ \sum_i z_i^2 = 0, $$ what restrictions does this place on the arrangement of the $z_i$ with respect to one another?

There is of course a rotational degree of freedom but I am gravitating toward the idea that the sum of squares being zero restricts the arrangement to one of a high degree of symmetry, relating to the kinds of polygons that can be constructed from $N$ points.

This seems like something that would have been well studied however I am struggling to find any material relating to it.

Answer 1

It's just one equation, so not a very high degree of symmetry. The squares $z_i^2$ form an arrangement of points on the circle whose centre of mass is at $0$. Given these, each $z_i$ has two possible values opposite each other on the circle.

Answer 2

The squaring really adds nothing to the question. You might just as well ask what such a set looks like if $\sum z_i=0$. And you can start with any set of $N-2$ numbers, as long as their sum has absolute value less than $2$; then you can pick the last two numbers to bring the sum to $0$. The only way that this can fail is if you are requiring that the numbers are all distinct; then you might find that one of your last two numbers is equal to one of the original $N-2$ numbers. But this is a "probability-zero" case.