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(From a problem in statistical physics) I'm trying to find the general properties of solutions (i.e., possible values of $t_0$) to the equation $$1 = \sum_{\ell=1}^{N} \frac{1}{x_{\ell} + t_0},$$ which can alternatively be written as the $N$th order polynomial $$ \prod_{\ell=1}^{N} (x_{\ell} + t_0) - \sum_{j=1}^{N} \sum_{i\neq j}^{N} (x_i + t_0) = 0$$ where $x_j\geq 0$ for all $j$. Because this equation can be written as an $N$th order polynomial, the fundamental theorem of algebra assures us that it has $N$ complex roots. That is, the two equations, can be reduced to \begin{equation} \prod^{N}_{m=1} \Big[ q_{m}(\{x_{\ell}\}) - t_0\Big] = 0, \end{equation} where $q_{m}(\{x_{\ell}\}) = q_{m}(x_1, \ldots, x_{N})$ are the (possibly complex) roots of the given polynomial.

My (more specific) question: If $x_{\ell} \geq 0$ for all $\ell$, what are the properties (if any) of the solutions $q_{\ell}(\{x_i\})$? For example, are there general conditions when they're exclusively real?

(Also will appreciate if someone can point me to a reference that deals with problems like this)

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    It's not too clear what general properties you are looking for, but may be worth noting that if $P(t)=\prod_{k=1}^N (t+x_k)$ then the given equation is equivalent to $P'(t_0) / P(t_0) = 1\,$. – 2017-01-02
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    Yes, I know that. The above condition was actually derived from minimizing $G(t) = - t+ \ln P(t)$. The properties I'm looking for are something like whether we could say if the roots are real or not. – 2017-01-03

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