Note on vanishing power sums of roots of unity

For a given positive integers $m$ and $\mathrm{\ell }$, we give a complete list of positive integers $n$ for which their exist $m$th roots of unity $x_1,\dots,x_n \in $\mathbb{C}$$suchthat$x_1^{\ell} + \cdots + x_n^{\ell}=0$. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers $n$ with $2\le n\le m$, there are distinct $m$th roots of unity $x_1,\dots ,x_n\in \mathbb{C}$ such that $x_1^{\mathrm{\ell }}+\cdots +x_n^{\mathrm{\ell }}=0$.