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System of nonlinear algebraic equations

  1. May 13, 2009 #1
    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 [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
     
  2. jcsd
  3. May 13, 2009 #2

    Hurkyl

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    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  4. May 13, 2009 #3
    Thanks Hurkyl, I've never thought about it that way. That helps a lot.
     
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