Classes of Holomorphic Multicomplex-Valued Functions Generated by Elliptic-Admissible Involutions

Abstract

We classify and count the real-algebra involutions of the multicomplex algebra $\mathbb{MC}(n)$ that map each element of its canonical monomial basis to a signed monomial, using a matrix model over ${\mathbb{F}}_2$. An \emph{elliptic-admissible pair} consists of such an involution $\sigma$ and a monomial unit $\mathbf{i}\in \mathbb{I}(n)$ satisfying ${\mathbf{i}}^2=-1$ and $\sigma (\mathbf{i})=-\mathbf{i}$. The fixed algebra of $\sigma$ is a real form for the complex structure ${\mathcal{L}}_{\mathbf{i}}$ defined by multiplication by $\mathbf{i}$, and the pair yields a Cauchy--Riemann system. We prove that its solution class depends only on $\mathbf{i}$, not on $\sigma$, and is exactly the class of mappings holomorphic with respect to ${\mathcal{L}}_{\mathbf{i}}$. Multicomplex holomorphy is recovered as the intersection of the classes associated with the elementary generators. We obtain the analogous characterization for anti-holomorphic classes, describe twisted systems intertwining two such complex structures, and prove that every $\mathbf{i}$-holomorphic or $\mathbf{i}$-anti-holomorphic mapping is componentwise harmonic (solutions to Laplace's equation).