A system of $2N$ equations resembling a Vandermonde matrix

Question

Suppose that I have a collection of known $\gamma_1, \dots, \gamma_{2N} \in \mathbb{C}$. Is there a known method to compute $\zeta_1, \dots, \zeta_N, \alpha_1, \dots, \alpha_N \in \mathbb{C}$ that solve the following system of equations?

$$ \begin{pmatrix} \zeta_1&\zeta_2&\dots&\zeta_N \\ \zeta_1^2&\zeta_2^2&\dots&\zeta_N^2 \\ \vdots&\vdots&\ddots&\vdots \\ \zeta_1^{2N}&\zeta_2^{2N}&\dots&\zeta_N^{2N} \end{pmatrix} \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_{N} \end{pmatrix} = \begin{pmatrix} \gamma_1 \\ \gamma_2 \\ \vdots \\ \gamma_{2N} \end{pmatrix} $$

Answer 1

Your system is $$\sum_{k=1}^N\alpha_k\zeta_k^j=\gamma_j, \quad 1\leq j\leq 2N.\quad\quad\quad(1)$$ Putting $x_k=\alpha_k\zeta_k$ we obtain $$\sum_{k=1}^Nx_k\zeta_k^j=\gamma_{j+1},\quad 0\leq j\leq 2N-1.\quad\quad\quad(2)$$
This system of equation was studied much, and it is called the Sylvester-Ramanujan system. A necessary and sufficient condition of solvability is given in the paper of Y. I. Lyubich,

The Sylvester-Ramanujan system of equations and the complex power moment problem. Ramanujan J. 8 (2004), no. 1, 23–45.

It is somewhat complicated to be reproduced here; they also give a uniqueness statement. Let me sketch Ramanujan's method of solving it. Consider the rational function $$f(z)=\sum_{k=1}^n\frac{x_k}{1-z\zeta_k},$$ then our system is equivalent to $$f(z)=\sum_{j=0}^{2N-1}\gamma_{j+1}z^j+O(z^{2N}),\quad z\to 0.$$ On the other hand, the rational function $f=A/B$ is a ratio of two polynomials of degree $N$. So the problem is to find two polynomials of degree $N$ whose ratio has prescribed $2N-1$ coefficients at $0$. This is not difficult, and nowadays this is called the "diagonal Pade approximation". The problem is also equivalent to the finite "moment problem", but the moment problem is usually studied when $x_k$ are positive while $\zeta_k$ and $\gamma_j$ are real.

Remark. The trouble may arise when some $\zeta_k$ are equal or some $\alpha_k$ are zero. Lyubich gives a (generic) necessary and sufficient condition on the RHS for these degenerations not to happen. For generic RHS, the system has unique solution.