# The Number of Zeros of Unilateral Polynomials over Coquaternions Revisited

## TL;DR

This paper provides a complete proof of the maximum number of zeros for unilateral coquaternionic polynomials, characterizes their zero-sets, and introduces an algorithm for zero classification with numerical examples.

## Contribution

It offers a new, simpler proof for the maximum zeros of coquaternionic polynomials and introduces an algorithm to compute and classify these zeros.

## Key findings

- Maximum of n(2n-1) zeros for degree n polynomials proven.
- Complete characterization of zero-sets provided.
- Algorithm for zero computation and classification developed.

## Abstract

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovsk\'a and Opfer [Electronic Transactions on Numerical Analysis, Volume 46, pp. 55-70, 2017], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree $n$ has, at most, $n(2n-1)$ zeros. In this paper we present a full proof of the referred result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero-sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.