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I came across a somewhat special system of nonlinear algebraic equations which I think must have been the subject of consideration in some book or article. I failed however to find such a resource, so I hope you can help out and point me somewhere.

The system consists of n equations and n unknowns [itex]x_1,\dots,x_n[/itex] and has the form

[tex]

\begin{align*}

c_1=&(-1)^{n}\left[x_1+\dots+x_n\right]\\

c_2=&(-1)^{n-1}\left[x_1x_2+\dots+x_1x_n+x_2x_3+\dots+x_{n-1}x_n\riight]\\

&\dots\\

c_n=&-x_1x_2\cdot\dots\cdot x_{n-1}x_n

\end{align*}

[/tex]

so that in the kth equation there is the sum of all possible products of k different x's. Has anybody seen this type of system before and know if it can be solved?

Thank you very much