The algebra of slice functions

Abstract

In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative ${}^{\ast }$-algebra $A$ over $\mathbb{R}$. These recently introduced function theories generalize to higher dimensions the classical theory of functions of a complex variable. Slice functions over $A$, which comprise all polynomials over $A$, form an alternative ${}^{\ast }$-algebra themselves when endowed with appropriate operations. We presently study this algebraic structure in detail and we confront with questions about the existence of multiplicative inverses. This study leads us to a detailed investigation of the zero sets of slice functions and of slice regular functions, which are of course of independent interest.

38 pages, to appear in Transactions of the American Mathematical Society

References in corpus (7)

• Continuous slice functional calculus in quaternionic Hilbert spaces
• Factorization of Rational Curves in the Study Quadric and Revolute Linkages
• Twistor transforms of quaternionic functions and orthogonal complex structures
• Singularities of slice regular functions over real alternative *-algebras
• Semigroups over real alternative *-algebras: generation theorems and spherical sectorial operators
• Power and spherical series over real alternative *-algebras
• Some properties for quaternionic slice-regular functions on domains without real points

Cited by in corpus (15)

• Slice regular functions in several variables
• Division algebras of slice functions
• Geometric function theory over quaternionic slice domains
• Cauchy-Riemann operators and local slice analysis over real alternative algebras
• On a class of orientation-preserving maps of ${\mathbb{R}}^4$
• Equivalence of slice semi-regular functions via Sylvester operators
• Almansi-type theorems for slice-regular functions on Clifford algebras
• Slice conformality and Riemann manifolds on quaternions and octonions
• Eigenvalue problems for slice functions
• A unified notion of regularity in one hypercomplex variable
• A new approach to slice analysis via slice topology
• Spherical coefficients of slice regular functions
• A unified theory of regular functions of a hypercomplex variable
• Quaternionic slice regularity beyond slice domains
• A local Cauchy integral formula for slice-regular functions