Check of reality for complex algebraic functions

Abstract

Consider a real algebraic curve with set of real points $R\neq\emptyset$ 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 $f(p)=f(p')$ for symmetrical points $p,p'\in D$. In particulary, this gives a proof of Boris and Michael Shapiro conjecture (2002).