Cauchy-Riemann equations and Jacobians of quaternion polynomials

TL;DR

This paper explores quaternion polynomial maps, deriving their Jacobian matrices, Cauchy-Riemann equations, and showing the Jacobian determinant is non-negative, extending concepts from complex analysis to quaternionic functions.

Contribution

It provides explicit formulas for Jacobians of quaternion polynomials and establishes non-negativity of their determinants, linking quaternionic analysis to classical complex theory.

Findings

Jacobian of quaternion polynomial maps computed explicitly

Cauchy-Riemann equations derived for quaternion functions

Jacobian determinant shown to be non-negative

Abstract

A map $f$ from the quaternion skew field $H$ to itself, can also be thought as a transformation $f:R^4 \to R^4$. In this manuscript, the Jacobian $J(f)$ of $f$ is computed, in the case where $f$ is a quaternion polynomial. As a consequence, the Cauchy-Riemman equations for $f$ are derived. It is also shown that the Jacobian determinant of $f$ is non negative over $H$. The above commensurates well with the theory of analytic functions of one complex variable.