On the real differential of a slice regular function
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Amedeo Altavilla
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.
Acknowledgements
The present work was developed while I was a PhD student in Mathematics at the University of Trento. For this reason I want to thank my former institution and especially my supervisor Prof. Alessandro Perotti. Furthermore I was partially supported by the Project FIRB “Geometria Differenziale e Teoria Geometrica delle Funzioni” and by GNSAGA of INdAM.
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
