On the real differential of a slice regular function

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 \textit{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.

23 pages, some adjustment in the structure of the sections, some typos removed, last example reviewed

References in corpus (4)

• Twistor transforms of quaternionic functions and orthogonal complex structures
• A new series expansion for slice regular functions
• Power and spherical series over real alternative *-algebras
• Some properties for quaternionic slice-regular functions on domains without real points

Cited by in corpus (7)

• Slice regularity and harmonicity on Clifford algebras
• Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in ${\mathbb{CP}}^3$
• On a class of orientation-preserving maps of ${\mathbb{R}}^4$
• Equivalence of slice semi-regular functions via Sylvester operators
• Slice regular functions as covering maps and global $\star$-roots
• Quaternionic slice regularity beyond slice domains
• Spherical coefficients of slice regular functions