Roots of sum of "inverse" monomials Consider the following equation (in variable $x$) for $\lambda_i\geq  0$
$$
p(x)=\frac{w_1}{(x-\lambda_1)}+\frac{w_2}{(x-\lambda_2)}\cdots+\frac{w_n}{(x-\lambda_n)}
$$
I am interested in characterizing the roots of the above equation. Is there any body of literature which I can refer to?
Specifically, I am interested bounding the sum of top $k$ roots of the above polynomial for the case when $\sum_{i=1}^nw_i^2 = 1$
 A: We may assume that the $\lambda_i$ are distinct and that $\lambda_i \lt \lambda_j$ for $i \lt j$. 
Given the $\lambda_i$ but no assumption on the $w_i,$ we can get the roots to be any set of $n-1$ points on the real line. In fact we can get it to be any set of $n-1$ or less distinct points except that the number is equal in parity to $n-1.$ So you need to be more specific. If the $w_i$ are all positive then the roots you seek interlace the $\lambda_i$ so the sum of the top $k$ roots is somewhere between $\sum_{n-k}^{n-1}\lambda_i$ and $\sum_{n-k+1}^{n}\lambda_i.$ These roots are also the places where $\prod|x-\lambda_i|^{m_i}$ has local maxima.


So perhaps for a reference you could look into interlacing of roots. However I don't have a good place for you to start.


The location of the roots is unchanged if we replace $p(x)$ with $cp(x)$ so we may assume $\sum w_i=1.$ The case $\sum w_i=0$ might be interesting , but I'll ignore it here.
The numerator of $$\sum_1^n\frac{w_i}{x-\lambda_i}=\frac{n(x)}{\prod(x-\lambda_i)}= \frac{x^{n-1}+\sum_0^{n-2}a_jx^j}{\prod(x-\lambda_i)} $$ is a monic polynomial of degree $n-1$.
Any choice of the $n-1$ values $w_1,w_2,\cdots,w_{n-1}$ (with $w_n=1-\sum_1^{n-1}w_i)$ uniquely determines the $n-1$ coefficients $a_0,a_1,\cdots,a_{n-2}.$ The converse is true as well, we can pick $n(x)$ to be any monic polynomial of degree $n-1$ and the $w_i$ are uniquely determined. It might be reasonable to specify that the $w_i$ are non-zero, but I won't in order to preserve this correspondence. From the way I specified the denominator, having a particular $w_i=0$ amounts to making $\lambda_i$ a root of $n(x).$ 
From now on I will assume that the $w_i$ are all positive and give two proofs that there is exactly one root in each interval $(\lambda_i,\lambda_{i+1})$ 
First proof: $p(x)$ is continuous in $(\lambda_{i},\lambda_{i+1}).$ Also, the one sided limits $$\lim_{x\to \lambda_i}p(x)=\pm\infty$$ with $+ \infty$ from above and $-\infty$ from below. This means the graph crosses the axis in each interval and there can only be $n-1$ roots.
Second proof: Consider $q(x)=\prod(|x-\lambda_i|)^{w_i}.$ It is continuous and non-negative touching the axis at each $\lambda_i$ but nowhere else. It has a local maximum  in each $(\lambda_{i},\lambda_{i+1}).$ Also,  $\ln |q(x)|$  has a local maximum at the same locations. Thus the derivative of $\ln|q(x)|$ has a zero in each interval (again at the same places). But that derivative is $p(x).$
A: Concerning the literature, Morris Marden's "Geometry of Polynomials" from 1966 is a good place to start reading, for instance, the first chapter gives a number of different possible interpretations of the sums you consider.
Kenneth B. Stolarsky's review in
The American Mathematical Monthly Vol. 112, No. 7 (Aug. - Sep., 2005), pp. 664-671
of two more recent books on the subject contains a lot of additional information and is a fine read, reviewed books:
Analytic Theory of Polynomials by Qazi Ibadur Rahman, Gerhard Schmeisser;
Complex Polynomials by Terry Sheil-Small.
From the introduction of Marden's text, you can easily see, that the case 
$$ \sum w_i \ = \ 0 \quad (*)$$
not explicitely adressed by Aaron Meyerowitz is an exceptional case you should take note of: Clearing denominators gives a polynomial on the right hand side, whose degree is lower than n-1 if and only if $(*)$ is true, so you will have less roots in that case. This is quite obvious in terms of physics, especially fluid dynamics:
The roots of $p(x)$ are the stagnation points of a flow which has sources and sinks, corresponding to positive and negative signs of the $w_i$ of strength $|w_i|$. If you have for example a unit source and a unit sink, there will be no stagnation points and hence no root, one less than expected, if $(*)$ does not vanish.
