  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
$\mathbb{CP}^3$ and we conclude by giving some qualitative results on the
complex structures induced by $\mathcal{C}$ via the twistor projection.
