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

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

2005-12-15

arxiv.org/abs/math/0512370v1

Notions of positivity and the geometry of polynomials, trends in mathematics, Springer, Basel, 2011, p. 167-178

Mathematics

Algebraic Geometry

21 pages

Scientific paper

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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