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On the zeros and critical points of a rational map. (English) Zbl 1007.30006

Let \(f:{\mathbf P}^1\to{\mathbf P}^1\) be a rational map, \(\{\alpha_i\}_{i\in I}\) the set of zeros of \(f\), \(\{w_j\}_{j\in J}\) the set of critical points of \(f\) which are not zeros of \(f\). The paper studies the relations between the points \(\alpha_i\) and the points \(w_j\), weighted with their multiplicities. Some geometrical interpretations of these relations are also given.

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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