Let $v\_{j}\in \mathbb{C}, 1\leq j\leq m$ and $w\in \mathbb{C}\setminus \{v\_{j}\}\_{j=1}^{m}$ and $n>0$.



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> Q: Can we say anything about the m roots $w\_{1},...,w\_{m}$ of
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$$p(w)=n+\sum\_{j=1}^{m}\frac{v\_{j}}{w-v\_{j}}=0?$$


Can we say anything about their approximate location in $\mathbb{C}\setminus \{v\_{j}\}\_{j=1}^{m}$ for a fixed collection of $\{v\_{j}\}\_{j=1}^{m}$. Does studying the roots of such functions fall under any framework? (incidentally, in case it helps to ring any bells, the antiderivative has a nice form
$$wn+log(\prod\_{j=1}^{m}(w-v\_{j})^{v\_{j}})+constants)$$


Do we have a formula for these roots? Do we know their location with respect to the constants $\{v\_{j}\}\_{j=1}^{m}$?



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> Q2: Supposing that $v\_j$ are distant enough from each other, are the roots $w\_j$ within some disjoint disks centered at each of the $v\_j$ eg. do we have $w\_{j}\in B\_{r\_{j}}(v\_{j})$ for some radii collection $(r\_{j})\_{j=1}^{m}$? Is there a way to use resultants techniques like mentioned in the answer below to help with this Q2?
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> Q3: Has that equation shown up in more recent times in some context? It is very reminiscent of equations from random matrix theory, Coulomb log-gases.
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This will be very helpful in a different question [No common roots of complex polynomial and of its derivative](https://mathoverflow.net/questions/362060/no-common-roots-of-complex-polynomial-and-of-its-derivative).


The motivation is studying Bethe roots for a particular system.


Update: just to be clear the question is still open. I think even just answering Q2 in the positive/negative will be very interesting.



Roots for $p(w)=n+\sum_{j=1}^{m}\frac{v_{j}}{w-v_{j}}$
As Alexandre said, there is not enough guidance to as to what kind of info about these equations the OP is after. In any case, something of interest for equations given in the above form is that there is a nice formula for the resultant of two such equations. It involves the *matrix-tree theorem*. The formula was [discovered](https://eudml.org/doc/147779) by Borchardt and then was [rediscovered](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s2-6.1.468) by A. L. Dixon.


