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Elementary proof of the B. and M. Shapiro conjecture for rational functions

Alexandre Eremenko·Andrei Gabrielov
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Abstract

We give a new elementary proof of the following theorem: if all critical points of a rational function g belong to the real line then there exists a fractional linear transformation L such that L(g) is a real rational function. Then we interpret the result in terms of Fuchsian differential equations whose general solution is a polynomial and in terms of electrostatics.

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Alexandre Eremenko, Andrei Gabrielov. 2005-12-15. Elementary proof of the B. and M. Shapiro conjecture for rational functions. https://arxiv.org/abs/math/0512370

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