Elementary proof of the B. and M. Shapiro conjecture for rational functions
Algebraic Geometry
2012-02-07 v1 Complex Variables
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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Cite
@article{arxiv.math/0512370,
title = {Elementary proof of the B. and M. Shapiro conjecture for rational functions},
author = {Alexandre Eremenko and Andrei Gabrielov},
journal= {arXiv preprint arXiv:math/0512370},
year = {2012}
}
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21 pages