Zeros of regular functions of quaternionic and octonionic variable: a division lemma and the camshaft effect

TL;DR

This paper investigates the zeros of regular functions over quaternions and octonions, establishing a division lemma, analyzing zero relations, and proving a fundamental theorem of algebra for octonionic polynomials.

Contribution

It introduces a division lemma for power series and provides a strong form of the fundamental theorem of algebra for octonionic polynomials, including zero multiplicity and factorization results.

Findings

Relation between zeros of octonionic regular functions and their products

Sum of zero multiplicities equals polynomial degree

Factorization into linear polynomials

Abstract

We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular functions and those of their product. In the case of octonionic polynomials, we get a strong form of the fundamental theorem of algebra. We prove that the sum of the multiplicities of zeros equals the degree of the polynomial and obtain a factorization in linear polynomials.