Suppose we have the complex numbers $x_1,x_2,\dots,x_N$ and that we know that
$$x_1+\dots+x_N =0,$$
$$x_1^2+\dots+x_N^2 =0,$$
$$\dots$$
$$x_1^N+\dots+x_N^N =0.$$
Does this imply that all $x_m=0$? There should be enough equations to get a unique solution. I have proven it in the case $N=2$, but going further requires a lot of algebra, and I'm wondering if it can be seen in an easier way than solving the system of equations by brute force.
Sum of powers of complex numbers equals zero implies the numbers themselves are zero
complex numbers
Best Answer
Yes. By the Newton-Girard formulas the fact that $p_1=p_2=\ldots=p_N=0$ implies the fact that $e_1=e_2=\ldots=e_N=0$, so the variables $x_1,\ldots,x_N$ are roots of the polynomial $z^N$, i.e. they are all zero.
A celebrated consequence is the following: if $M$ is a $N\times N$ real symmetric matrix such that $\text{Tr}(M)=\text{Tr}(M^2)=\ldots=\text{Tr}(M^N)=0$, then $M=0$.