Separating Morphisms from Real Algebraic Curves

Given a real algebraic curve we consider the set of all morphisms to the projective line with the property that the preimage of every real point consists entirely of real points. It turns out that this generalises the notion of interlacing polynomials on the real line to projective curves. Using this theory, we will answer a question raised by Shamovich and Vinnikov on hyperbolic curves as well as a question by Fiedler-LeTouzé on totally real pencils on plane curves. This is joint work with Kristin Shaw.