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After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular functions, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function f : Ω → H a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on Ω (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent f better and better and allow to prove increasingly stronger results. [ABSTRACT FROM AUTHOR]

The concept of slice regular function over the real algebra H of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let Ω ⊂ H be a domain, i.e., a non-empty connected open subset of H = R 4 . Suppose that Ω intersects R and is invariant under rotations of H around R . A function f : Ω → H is slice regular if it is of class C 1 and, for all complex planes C I spanned by 1 and a quaternionic imaginary unit I ( C I is a 'complex slice' of H ), the restriction f I of f to Ω I = Ω ∩ C I satisfies the Cauchy–Riemann equations associated with I, i.e., ∂ ¯ I f I = 0 on Ω I , where ∂ ¯ I = 1 2 (∂ ∂ α + I ∂ ∂ β ) . Given any positive natural number n, a function f : Ω → H is called slice polyanalytic of order n if it is of class C n and ∂ ¯ I n f I = 0 on Ω I for all I. We define global slice polyanalytic functions of order n as the functions f : Ω → H , which admit a decomposition of the form f (x) = ∑ h = 0 n - 1 x ¯ h f h (x) for some slice regular functions f 0 , ... , f n - 1 . Global slice polyanalytic functions of any order n are slice polyanalytic of the same order n. The converse is not true: for each n ≥ 2 , we give examples of slice polyanalytic functions of order n, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations ∂ ¯ I n f I = 0 on Ω I and as solutions of their global version ϑ ¯ n f = 0 on Ω \ R . Our quaternionic results extend to the slice monogenic case. [ABSTRACT FROM AUTHOR]

In this paper, we lay the foundations of the theory of slice regular functions in several (non-commuting) variables ranging in any real alternative ∗ -algebra, including quaternions, octonions and Clifford algebras. This higher dimensional function theory is an extension of the classical theory of holomorphic functions of several complex variables. It is based on the construction of a family of commuting complex structures on R 2 n . One of the relevant aspects of the theory is the validity of a Cauchy-type integral formula and the existence of ordered power series expansions. The theory includes all polynomials and power series with ordered variables and right coefficients in the algebra. We study the real dimension of the zero set of polynomials in the quaternionic and octonionic cases and give some results about the zero set of polynomials with Clifford coefficients. In particular, we show that a nonconstant polynomial always has a non empty zero set. [ABSTRACT FROM AUTHOR]

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra O , recently introduced by M. Jin, G. Ren and I. Sabadini. A function f : Ω D ⊂ O → O is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra H I of O generated by a pair I = (I , J) of orthogonal imaginary units I and J ( H I is a 'quaternionic slice' of O ), the restriction of f to H I belongs to the kernel of the corresponding Cauchy–Riemann–Fueter operator ∂ ∂ x 0 + I ∂ ∂ x 1 + J ∂ ∂ x 2 + (I J) ∂ ∂ x 3 . The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their 'holomorphic nature': slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine 8-dimesional domains of O . Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator Γ and of slice Fueter operator ϑ ¯ F over octonions, which allow to characterize the slice Fueter-regular functions as the C 2 -functions in the kernel of ϑ ¯ F satisfying a second order differential system associated with Γ . [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian matrix J f of a slice regular function f proving in particular that det (J f) ≥ 0 , i.e., f is orientation-preserving. We give a complete characterization of the fibers of f making use of a new notion we introduce here, the one of wing of f. We investigate the singular set N f of f, i.e., the set in which J f is singular. The singular set N f turns out to be equal to the branch set of f, i.e., the set of points y such that f is not a homeomorphism locally at y. We establish the quasi-openness properties of f. As a consequence we deduce the validity of the Maximum Modulus Principle for f in its full generality. Our results are sharp as we show by explicit examples. [ABSTRACT FROM AUTHOR]

We present an efficient algorithm for the construction of a basis of H2(Ω¯,∂Ω;ℤ) via the Poincaré-Lefschetz duality theorem. Denoting by g the first Betti number of Ω¯ the idea is to find, first g different 1-boundaries of Ω¯ with supports contained in ∂Ω whose homology classes in ℝ3∖Ω form a basis of H1(ℝ3∖Ω;ℤ), and then to construct a set of 2-chains in Ω¯ having these 1-boundaries as their boundaries. The Poincaré-Lefschetz duality theorem ensures that the relative homology classes of these 2-chains in Ω¯ modulo ∂Ω form a basis of H2(Ω¯,∂Ω;ℤ). We devise a simple procedure for the construction of the required set of 1-boundaries of Ω¯ that, combined with a fast algorithm for the construction of 2-chains with prescribed boundary, allows the efficient computation of a basis of H2(Ω¯,∂Ω;ℤ) via this very natural approach. Some numerical experiments show the efficiency of the method and its performance comparing with other algorithms. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide and prove the most general Cauchy integral formula for slice regular functions and for $$C^1$$ functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the existence of algebraic models for Nash sets having suitable symmetries. Given a Nash set $$M \subset {\mathbb {R}}^n$$ , we say that $$M$$ is specular if it is symmetric with respect to an affine subspace $$L$$ of $${\mathbb {R}}^n$$ and $$M \cap L=\emptyset $$ . If $$M$$ is symmetric with respect to a point of $${\mathbb {R}}^n$$ , we call $$M$$ centrally symmetric. We prove that every specular compact Nash set is Nash isomorphic to a specular real algebraic set and every specular noncompact Nash set is semialgebraically homeomorphic to a specular real algebraic set. The same is true replacing 'specular' with 'centrally symmetric', provided the Nash set we consider is equal to the union of connected components of a real algebraic set. Less accurate results hold when such a union is symmetric with respect to a plane of positive dimension and it intersects that plane. The algebraic models for symmetric Nash sets $$M$$ we construct are symmetric. If the local semialgebraic dimension of $$M$$ is constant and positive, then we are able to prove that the set of birationally nonisomorphic symmetric algebraic models for $$M$$ has the power of continuum. [ABSTRACT FROM AUTHOR]