Check of reality for complex algebraic functions

Abstract

Consider a real algebraic curve with set of real points $R\ne \varnothing$ and complexification $P\supset R$. Let $f$ be an algebraic function on $P$ with devisor of critical points $D\subset P$. We prove that $f$ is real after a linear-factorial transformation, if $D$ is symmetrical with respect to $R$ and $f(p)=f(p^{\prime })$ for symmetrical points $p,p^{\prime }\in D$. In particulary, this gives a proof of Boris and Michael Shapiro conjecture (2002).

This paper has been withdrawn by the author. Article removed, because the author has found serious deficiencies in the proof of the main theorem