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.
CITATION STYLE
Eremenko, A., & Gabrielov, A. (2011). An Elementary Proof of the B. and M. Shapiro Conjecture for Rational Functions. In Notions of Positivity and the Geometry of Polynomials (pp. 167–178). Springer Basel. https://doi.org/10.1007/978-3-0348-0142-3_10
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