Suppose that, given constants $\mu_0,...\mu_{n-1}$, there exists a solution $(x_1,...,x_n) \in \mathbb{R}_{+}^{n}$ to the following system :
$$\mu_i = \sum\limits_{j=1}^{n} x_j^i$$
Can we find it for any n ?
In the over way around, this is related to Vandermonde matrices. I tried some optimisation algorithms on it and it converges quite well, so I'm pretty sure there exist an analytical solution.
E.g, for $n=2$, take $x_{1,2} = \mu_1 \pm \sqrt{\frac{\mu_2}{2} - \mu_1^2}$
I am pretty sure that there is no analytical solution, as the system involves polynomials of degree up to $n$.
In the case $n=3$, the equations define a plane, a sphere and a cubic surface.
The intersection of the plane and the sphere is a circle. The intersection of the plane and the cubic surface can be three straight lines forming an equilateral triangle, or a set of curves asymptotic to these. There can be six intersections with the circle, and if I am right, you can obtain them by solving a sextic polynomial.
As $n$ grows, this quickly becomes intractable.