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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 functionsnaturally emerge in the twistor interpretation of slice regularity introducedin \cite{gensalsto} and developed in \cite{AAtwistor}. To any slice-polynomialfunction $P$ we associate its {\em companion} $P^\vee$ and its {\em extension}to the real axis $P_\mathbb{R}$, that are quaternionic functions naturallyrelated to $P$. Then, using the theory of twistor spaces, we are able to showthat 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 notionof degree. With the brand new tool of slice-polynomial functions, we {compute}the twistor discriminant locus of a cubic scroll $\mathcal{C}$ in$\mathbb{CP}^3$ and we conclude by giving some qualitative results on thecomplex structures induced by $\mathcal{C}$ via the twistor projection.

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