# System of nonlinear algebraic equations

• Pere Callahan

#### 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

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.

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