Slice regular functions on real alternative algebras

Abstract

In this paper we develop a theory of slice regular functions on a real alternative algebra $A$. Our approach is based on a well--known Fueter's construction. Two recent function theories can be included in our general theory: the one of slice regular functions of a quaternionic or octonionic variable and the theory of slice monogenic functions of a Clifford variable. Our approach permits to extend the range of these function theories and to obtain new results. In particular, we get a strong form of the fundamental theorem of algebra for an ample class of polynomials with coefficients in $A$ and we prove a Cauchy integral formula for slice functions of class $C^1$.

32 pages - accepted for publication in Advances in Mathematics

Cited by in corpus (89)

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• Volume Cauchy formulas for slice functions on real associative *-algebras
• Some properties for quaternionic slice-regular functions on domains without real points
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• Slice-Polynomial Functions and Twistor Geometry of Ruled Surfaces in ${\mathbb{CP}}^3$
• Cauchy-Riemann operators and local slice analysis over real alternative algebras
• The harmonicity of slice regular functions
• Manifolds over Cayley-Dickson algebras and their immersions
• On the real differential of a slice regular function
• On a quaternionic Picard theorem
• On a class of orientation-preserving maps of ${\mathbb{R}}^4$
• Almansi-type theorems for slice-regular functions on Clifford algebras
• The quaternionic Gauss-Lucas Theorem
• S-regular functions which preserve a complex slice
• Equivalence of slice semi-regular functions via Sylvester operators
• Landau's theorem for slice regular functions on the quaternionic unit ball
• The Growth and Distortion Theorems for Slice Monogenic Functions
• Generalized partial-slice monogenic functions
• A local representation formula for quaternionic slice regular functions
• Zeros of slice functions and polynomials over dual quaternions
• S--polyregular Bargmann spaces
• Slice conformality and Riemann manifolds on quaternions and octonions
• Extension theorem and representation formula in non-axially symmetric domains for slice regular functions
• A representation formula for slice regular functions over slice-cones in several variables
• Slice starlike functions over quaternions
• Slice regular functions and orthogonal complex structures over ${\mathbb{R}}^8$
• On power series expansions of the S-resolvent operator and the Taylor formula
• Eigenvalue problems for slice functions
• $\ast$-Logarithm for Slice Regular Functions
• Algebraic approach to slice monogenic functions
• A unified notion of regularity in one hypercomplex variable
• The orthogonal projection on slice functions on the quaternionic sphere
• Quaternionic Hankel operators and approximation by slice regular functions
• A new approach to slice analysis via slice topology
• A four dimensional Jensen formula
• Partial slice regularity and Fueter's Theorem in several quaternionic variables
• The Cauchy transform in the slice hyperholomorphic setting and related topics
• On a definition of logarithm of quaternionic functions
• Lagrange polynomials over Clifford numbers
• Almansi-type decomposition for slice regular functions of several quaternionic variables
• Some notions of subharmonicity over the quaternions
• Slice regular holomorphic Cliffordian functions of order $k$
• Implementing zonal harmonics with the Fueter principle
• Slice regular functions as covering maps and global $\star$-roots
• The twistor space of a real associative algebra: incidence geometry, semisimple classification and slice-regular functions
• Functions of the infinitesimal generator of a strongly continuous quaternionic group
• Quaternionic slice regularity beyond slice domains
• Spherical coefficients of slice regular functions
• Weak slice regular functions on the $n$-dimensional quadratic cone of octonions
• Slice Dirac operator over octonions
• A four dimensional Bernstein Theorem
• Riemann slice-domains over quaternions I
• Dunkl approach to slice regular functions
• A unified theory of regular functions of a hypercomplex variable
• A direct approach to quaternionic manifolds
• Monogenic functions over real alternative *-algebras: the several hypercomplex variables case
• A manifold Fueter-Sce phenomenon in one hypercomplex variable
• On geometric aspects of quaternionic and octonionic slice regular functions
• On the first-order theories of quaternions and octonions
• Adaptive Fourier decomposition of slice regular functions
• Julia theory for slice regular functions
• Growth theorems in slice analysis of several variables
• Schur functions and their realizations in the slice hyperholomorphic setting
• Boundary Julia theory for slice regular functions
• On the Quadratic Cone of ${\mathbb{R}}_3.$
• Noncommutative Cauchy integral formula
• Quaternionic slice regular functions and quaternionic Laplace transforms
• On slice polyanalytic functions of a quaternionic variable
• Function Theories in Cayley-Dickson algebras and Number Theory
• Ideals of regular functions of a quaternionic variable
• A local Cauchy integral formula for slice-regular functions
• Slice regular functions of several octonionic variables
• On Some Geometric Properties of Slice Regular Functions of a Quaternion Variable
• Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators
• Slice regular composition operators
• Quaternionic Cartan coverings and applications