0

Let $\mathbf{z}=(z_1,\cdots,z_n)$ be a point in $\mathbb{C}^n$. Assume that $z_j=x_j+\iota y_j$. Let $d\ge 3$. We want to maximize $x_1^2$ such that

\begin{align} & \sum_{j=1}^n z_j^d=0, \\ & \sum_{j=1}^n|z_j|^2=1. \end{align}

Note that if $d$ is odd, then the maximum value is $\frac{1}{2}$.

As we know if $\sum_{j=1}^n z_j^d=0$ and $d$ is odd, then $z_l+z_k=0$ for some $l$ and $k$ (see here).

Now I am stuck what happens if $d$ is even. Any hint will be appreciated. Thanks.

Sachchidanand Prasad
  • 2,890
  • 2
    $z_1^3+z_2^3=0$ doesn't imply $z_1+z_2=0$ (see $z_1=1, z_2=e^{i\pi/3}$); the result you quote is if the sum of odd powers is zero for all the odd powers, not for a specific one – Conrad Jan 27 '22 at 17:48
  • 1
    what you want to do is use that $x_1^2 \le |z_1|^2 \le (\sum_{k \ge 2}|z_k|^d)^{2/d} \le \sum_{k \ge 2}|z_k|^2$ (here one needs $d \ge 2$) to conclude the maximum is $1/2$ attained for $z_1=1/\sqrt 2, z_2=\alpha/\sqrt 2,z_3=z_4=..=z_n=0, \alpha^d=-1$ so for $d$ odd you can take $z_2=-z_1$ indeed – Conrad Jan 27 '22 at 17:59
  • @Conrad, you are right. Now I realized it. Thanks for your help. – Sachchidanand Prasad Jan 27 '22 at 19:03

0 Answers0