Holomorphic functions and regular quaternionic functions on the hyperkähler space H

Let H be the space of quaternions, with its standard hypercomplex structure. Let $\mathcal R(\Omega)$ be the module of \emph{$\psi$-regular} functions on $\Omega$. For every $p\in H$, $p^2=-1$, $\mathcal R(\Omega)$ contains the space of holomorphic functions w.r.t. the complex structure $J_p$ induced by $p$. We prove the existence, on any bounded domain $\Omega$, of $\psi$-regular functions that are not $J_p$-holomorphic for any $p$. Our starting point is a result of Chen and Li concerning maps between hyperk\"ahler manifolds, where a similar result is obtained for a less restricted class of quaternionic maps. We give a criterion, based on the energy-minimizing property of holomorphic maps, that distinguishes $J_p$-holomorphic functions among $\psi$-regular functions

An application of biregularity to quaternionic Lagrange interpolation

We revisit the concept of totally analytic variable of one quaternionic variable introduced by Delanghe \cite Delanghe} and its application to Lagrange interpolation by G\"uerlebeck and Spr\"ossig \cite{GS}. We consider left-regular functions in the kernel of the Cauchy-Riemann operator $$\mathcal D=2\left(\frac{\partial}{\partial{\bar z_1}}+j\frac{\partial}{\partial{\bar z_2}}\right)=\frac{\partial}{\partial{x_0}}+i\frac{\partial}{\partial{x_1}}+j\frac{\partial}{\partial{x_2}}-k\frac{\partial}{\partial{x_3}}.$$ For every imaginary unit p\in {\Sp}^2, let {\CC}_p=\langle 1,p\rangle\simeq {\CC} and let $J_p=p_1J_1+p_2J_2+p_3J_3$ be the corresponding complex structure on {\HH}. We identify totally regular variables with real--affine holomorphic functions from ({\HH},J_p) to ({\CC}_p,L_p), where $L_p$ is the complex structure defined by left multiplication by $p$. We then show that every $J_p$--biholomorphic map, which is always a biregular function, gives rise to a Lagrange interpolation formula at any set of distinct points in {\HH}. Publisher version at: http://link.aip.org/link/?APCPCS/1048/691/

A four dimensional Bernstein Theorem

We prove a four dimensional version of the Bernstein Theorem, with complex polynomials being replaced by quaternionic polynomials. We deduce from the theorem a quaternionic Bernstein's inequality and give a formulation of this last result in terms of four-dimensional zonal harmonics and Gegenbauer polynomials

The quaternionic Gauss-Lucas Theorem

The classic Gauss-Lucas Theorem for complex polynomials of degree $d\ge2$ has a natural reformulation over quaternions, obtained via rotation around the real axis. We prove that such a reformulation is true only for $d=2$. We present a new quaternionic version of the Gauss-Lucas Theorem valid for all $d\geq2$, together with some consequences.Comment: 7 pages, 1 figure. Remarks added in section 3. Proposition 14 added with complete proo

Slice regular functions on real alternative algebras

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 $C^1$