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I have the following equation: $$\sum_{k=0}^n \frac{a_k}{a_k+x}=1$$ where all the $a_k$'s are positive real numbers.

For $n=2$ the roots are $x={}_{-}^+\sqrt{a_1a_2}$, but for $n\geq 3$ the expression for $x$ seems to become very messy.

So my question is, how to proceed solving for $x$, or at least try to understand the nature of the roots of this equation for $n\geq 3$?

$\mathbf{Remark}$: This equation comes in connection to the problem of assigning optimal powers to the BTSs in a CDMA cell. There $x$ corresponds to the Perron-Frobenius eigenvalue of a matrix constructed with the channel gains between BTSs and the mobile stations.

Samrat Mukhopadhyay
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1 Answers1

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For each $n\geqslant1$, this is equivalent to $P(x)=0$, where $P$ is a monic polynomial of degree $n$ whose coefficients are functions of $(a_k)_{1\leqslant k\leqslant n}$. If $n\leqslant4$, one can solve algebraically for the roots of $P$. If $n\geqslant5$, one can solve numerically for the roots of $P$. Note additionally that, if $n\geqslant2$, there is always exactly one positive real root.

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