On the Quadratic Cone of $\mathbb{R}_3.$

TL;DR

This paper explores functions on the quadratic cone of the algebra , introducing a slice regular theory, discussing a Cauchy formula, analyzing zeros, and deriving a determinant formula for matrices over this cone.

Contribution

It introduces a slice regular function theory on the quadratic cone of  and provides new formulas for zeros and determinants within this algebraic structure.

Findings

Established a representation of the quadratic cone using quaternionic decomposition.

Developed a slice regular theory and a Cauchy formula for functions on the cone.

Derived a formula for the determinant of matrices with entries in the quadratic cone.

Abstract

In this paper we study the following type of functions $f: \mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $ \mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra $\mathbb{R}_{3}$. From the fact that it is possible to write the algebra $ \mathbb{R}_{3}$ as a direct sum of quaternions, we get the observation that it is possible to find a clever representation for $ \mathcal{Q}_{\mathbb{R}_3}$. By using this result a slice regular theory was introduced and a Cauchy formula is discussed. Moreover, a detailed study of the zeros is performed. Finally, we find a formula for the determinant of a matrix with entries in $ \mathcal{Q}_{\mathbb{R}_3}$.