Test for reality of algebraic functions

TL;DR

This paper establishes a criterion for when complex algebraic functions on real algebraic curves are equivalent to real algebraic functions, based on the stability of the divisor of preimages of critical values under complex conjugation.

Contribution

It provides a necessary and sufficient condition involving divisor stability for the reality of algebraic functions on real algebraic curves.

Findings

Proves the equivalence condition involving divisor stability.

Characterizes when complex algebraic functions are real.

Connects divisor properties with function reality.

Abstract

In this paper is proved that a complex algebraic function on complexification of a real algebraic curve is equivalent to real algebraic function, if and only if the divisor of preimage of critical values is stable under the involution of complex conjugation.