Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in ${\mathbb{C}\mathbb{P}}^3$

In the present paper we introduce the class of slice-polynomial functions: slice regular functions {defined over the quaternions, outside the real axis,} whose restriction to any complex half-plane is a polynomial. These functions naturally emerge in the twistor interpretation of slice regularity introduced in \cite{gensalsto} and developed in \cite{AAtwistor}. To any slice-polynomial function $P$ we associate its {\em companion} $P^{\vee }$ and its {\em extension} to the real axis $P_{\mathbb{R}}$, that are quaternionic functions naturally related to $P$. Then, using the theory of twistor spaces, we are able to show that for any quaternion $q$ the {cardinality of simultaneous} pre-images of $q$ via $P$, $P^{\vee }$ and $P_{\mathbb{R}}$ is generically constant, giving a notion of degree. With the brand new tool of slice-polynomial functions, we {compute} the twistor discriminant locus of a cubic scroll $\mathcal{C}$ in ${\mathbb{C}\mathbb{P}}^3$ and we conclude by giving some qualitative results on the complex structures induced by $\mathcal{C}$ via the twistor projection.