I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex numbers and the $p_j$'s are distinct complex numbers.

Any relevant reference is welcome.

Note : The Gauss-Lucas theorem implies that if the $a_j$'s are all assumed to be positive, then the roots of $R$ lie in the convex hull of the set of poles $\{p_1,\dots, p_n\}$. However, this is false in the general case, but perhaps there is something to be said.

Thank you, Malik


Section 21 in Chapter V of "Geometry of Polynomials" by Morris Marden, AMS, Mathematical Surveys 3, 1966 may contain what you are looking for, Lemma 21.1.on page 98 is as follows:

Let $C_j(z) := |z - c_j|^2 - r_j^2$ and $C_j$ be the circular region defined by $\pm C_j(z) \leq 0$, let $p_j$ be in $C_j$ and let $a_j$ be real or complex constants, then every root $Z$ of

$$ \sum_{j=1}^n \frac{a_j}{z - p_j} = 0 $$

is either contained in one of the $C_j$ or satisfies the inequality

$$ \left| \sum_{j=1}^n \frac{a_j(c_j - Z)}{C_j(Z)} \right|^2 - \left( \sum_{j=1}^n \frac{|a_j|r_j}{|C_j(Z)|} \right)^2 \ \leq \ 0 \, .$$

While it is not explicitly stated there, I suppose the $c_j$ are arbitrary complex numbers, while the $r_j$ are positive real numbers. I adapted the notation to your question and fixed what I considered a minor typo, but did no thorough proofreading.

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  • $\begingroup$ Thanks for the reference. It seems difficult though to obtain a geometrical interpretation from this lemma. For what I want, I think Theorem 8.2, p.32, is more interesting. $\endgroup$ – Malik Younsi Apr 17 '14 at 14:11
  • $\begingroup$ @Malik Younsi: You are welcome, I only cited the Lemma as it fitted your sum. I would study the whole book if I still had the time ... $\endgroup$ – thomashennecke Apr 17 '14 at 14:34

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