Boris SHAPIRO

Are rational functions with real critical points real?

ABSTRACT

Given a rational function in 1 variable of the form ϕ(x) = \frac{x^{n}+a_{2}x^{n−2}+...+a_{n}}{x^{n−1}+b_{2}x^{n−2}+...+b_{n}} we claim that if all its critical points are real then all the coefficients a_{i} and b_{i} are real. This result is a special case of more general conjecture on total reality in Schubert calculus and has an interesting reformulation in terms of all solutions to a certain simply looking system of 2nd order equations being real. No special knowledge is required and I am going to formulate a number of other (appealing) conjectures.

References [EG] A.Eremenko and B. Gabrielov, Rational functions with real critical points and

B. and M. Shapiro conjecture in real enumerative geometry, preprint

[BEGS] M.Bonk, A.Eremenko, B. Gabrielov and B.Shapiro Rational function with real critical points is equivalent to a real function, preprint