How hard is it to solve a system of equalities of the form

$a_{k1}x_1^k + \cdots + a_{kn}x_n^k = b_k$

with $k$ ranging from $1$ to $m$? I realize that this is a non-convex system but it seems plausible that it might be tractable. If the theoretical complexity is bad, how might one go about finding a feasible solution to such a system in practice? In my case I have $m < n \leq 10$. We also happen to know that $x_i \geq 0$, in case that helps. Other suggestions for tags are welcome.