I am thinking about this problem:
Let $\sum_{i=1}^N a_ix_i^k = b_k,\ k = 0,1,2,\dots,$ be an equation system. If I only know all the values of $b_k$'s, is there any way to find out the values of $a_i$'s and $x_i$'s for $i=1,2,\dots,N$? And how many equations do I need to use to solve $a_i$'s and $x_i$'s?
In the case that all the $a_i = 1$, I know that I can use Newton's equality and then solve the roots of a polynomial to find out $x_i$'s. But in the case that $a_i\neq 1$, I have no idea.
Thanks in advance for any advice you can offer.