Suppose $n$ and $N$ be positive integers $\geq 2$. Suppose $x_1, \cdots, x_n \in \mathbb{C}$ satisfies the following system ($\star$) of equations: \begin{align*} x_1+ &\cdots + x_n =0 \\ x_1^2+ &\cdots + x_n^2 =0 \\ &\cdots \\ x_1^N+ &\cdots + x_n^N =0 \end{align*} That is, $p_k(x_1, \cdots, x_n)=0$ for all $1 \leq k \leq N$, where $p_k(x_1, \cdots, x_n)= \sum_{i=0}^{n} x_i^k$ denotes the $k$-th power sum. Recall that Newton's identities say $$ ke_k(x_1, \cdots, x_n)=\sum_{i=1}^{k}(-1)^{i+1}e_{k-i}(x_1, \cdots, x_n)p_i(x_1, \cdots, x_n) $$ for all $n \geq k \geq 1$.
If $N \geq n$, then the solution of the system satisfies $e_k(x_1, \cdots, x_n)=0$ for all $k = 1, \cdots, n$. Thus $x_1= \cdots = x_n=0$, as they are the solutions of the equation $t^n=0$.
How about the case $N<n$? I believe the system ($\star$) has nontrivial solution because there are more variables than the number of equations, but I'm not able to verify it.
Question 1
For any $2 \leq N < n$, does there exists a solution $(x_1, \cdots, x_n) \in \mathbb{\left(C^{\times}\right)}^n $ for the system ($\star$)?
Define $S_\alpha =\{ re^{i\theta} \in \mathbb{C} : r>0, |\theta|\leq \alpha \}$ for $\alpha \in (0, \pi]$.
It is easy to see the system ($\star$) has no solution in $\left(S_\alpha\right)^n \subset \mathbb{\left(C^{\times}\right)}^n$, provided that $0 < \alpha \leq \frac{\pi}{2}$. But after direct calculation for small $n$ and $N$, I suspect that the following question has a positive answer, since we can let $n$ sufficiently larger.
Question 2
Does the system ($\star$) have a solution in $\left(S_\alpha\right)^n$ for every $0 < \alpha \leq \pi $?