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### A system of $2N$ equations resembling a Vandermonde matrix
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-...