System of nonlinear algebraic equations

[Pere Callahan]

Hello,

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 $x_1,\dots,x_n$ and has the form
$$\begin{align*} c_1=&(-1)^{n}\left[x_1+\dots+x_n\right]\\ c_2=&(-1)^{n-1}\left[x_1x_2+\dots+x_1x_n+x_2x_3+\dots+x_{n-1}x_n\riight]\\ &\dots\\ c_n=&-x_1x_2\cdot\dots\cdot x_{n-1}x_n \end{align*}$$

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

 

[Hurkyl]

Your right hand sides are essentially the elementary symmetric polynomials. Your system of equations is nothing more than asking to find the roots of a certain (monovariate) polynomial -- see Wikipedia.

 

[Pere Callahan]

Thanks Hurkyl, I've never thought about it that way. That helps a lot.