The theory of slice regular functions over the quaternions, introduced by Gentili and Struppa in [7], was born on domains that intersect the real axis. This hypothesis can be overcome using the theory of stem functions introduced by Ghiloni and Perotti ([8]), in the context of real alternative algebras. In this paper I will recall the notion and the main properties of stem functions. After that I will introduce the class of slice regular functions induced by stem functions and, in this set, I will extend the identity principle, the maximum and minimum modulus principles and the open mapping theorem. Differences will be shown between the case when the domain does or does not intersect the real axis.

Abstract. Given a slice regular function f : Ω ⊂ H → H, with Ω ∩ R = ∅, it is possible to lift it to a surface in the twistor space CP 3 of S 4 ≃ H ∪ {∞} (see [14]). In this paper we show that the same result is true if one removes the hypothesis Ω ∩ R = ∅ on the domain of the function f . Moreover we find that if a surface S ⊂ CP 3 contains the image of the twistor lift of a slice regular function, then S has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in CP 3 that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in Gr 2 (C 4 ) which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside Gr 2 (C 4 ), showing the role of slice regular functions not defined on R. At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions.

We give quantitative and qualitative results on the family of surfaces in CP 3 containing finitely many twistor lines. We start by analyzing the ideal sheaf of a finite set of disjoint lines E. We prove that its general element is a smooth surface containing E and no other line. Afterwards we prove that twistor lines are Zariski dense in the Grassmannian Gr(2, 4). Then, for any degree d ≥ 4, we give lower bounds on the maximum number of twistor lines contained in a degree d surface. The smooth and singular cases are studied as well as the j-invariant one.

According to [5] we define the * -exponential of a slice-regular function, which can be seen as a generalization of the complex exponential to quaternions. Explicit formulas for exp * (f ) are provided, also in terms of suitable sine and cosine functions. We completely classify under which conditions the * -exponential of a function is either slice-preserving or CJ -preserving for some J ∈ S and show that exp * (f ) is never-vanishing. Sharp necessary and sufficient conditions are given in order that exp *

Abstract:In this paper we show that the real differential of any injective slice regular function is everywhere invertible. The result is a generalization of a theorem proved by G. Gentili, S. Salamon and C. Stoppato and it is obtained thanks, in particular, to some new information regarding the first coefficients of a certain polynomial expansion for slice regular functions (called spherical expansion), and to a new general result which says that the slice derivative of any injective slice regular function is different from zero. A useful tool proven in this paper is a new formula that relates slice and spherical derivatives of a slice regular function. Given a slice regular function, part of its singular set is described as the union of surfaces on which it results to be constant.