All critical points of a rational function f are real only if f is equivalent to real

You need to know: Complex numbers, functions in complex variable, complex differentiation, polynomials.

Background: A (complex) rational function f is a ratio of two polynomials with complex coefficients, . If all coefficients of P(z) and Q(z) are real, we say that f is a real rational function. A complex number is a critical point of f if . We say that two rational functions f and g are equivalent if for some complex numbers a,b,c,d such that .

The Theorem: On 25th January 2000 Alexandre Eremenko and Andrei Gabrielov submitted to Annals of Mathematics a paper in which they proved, among other results, that if all critical points of a rational function f are real, then f is equivalent to a real rational function.

Short context: The Theorem equivalently states that, if for polynomials all the solutions of the equation are real, then there exist complex numbers such that and and are real polynomials. This is a special case of a well-known conjecture of B. and M. Shapiro, made around 1993, which predicts a similar result for any number of polynomials. In a later work, Mukhin, Tarasov, and Varchenko proved this conjecture in general.

Links: Free arxiv version of the original paper is here, journal version is here. See also Section 2.1 of this book for an accessible description of the Theorem.

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