A Note on Systems of Equations of Power Sums

Symmetric functions play an important role in several subjects of mathematics, such as algebraic combinatorics, representation theory of finite groups and algebraic geometry. In this paper, we study the solutions of the following system of equations related to bases of symmetric functions:
$$ \begin{cases} x_{1}^{k_1}+x_{2}^{k_1}+\cdots +x_{n}^{k_1}=a, \\ x_{1}^{k_2}+x_{2}^{k_2}+\cdots +x_{n}^{k_2}=a, \\ \vdots \\ x_{1}^{k_n}+x_{2}^{k_n}+\cdots +x_{n}^{k_n}=a, \end{cases} $$
where $0<k_1<k_2<\cdots<k_n$, $k_1,k_2,\cdots,k_n$ are natural numbers, and $a\in\{0,1,n\}$. Our main theorems generalize several known results.