Vieta's Formulas for Quaternionic Polynomials

TL;DR

This work extends Vieta-type relations to quaternionic polynomials by leveraging a basic polynomial construction to connect complex roots with quaternionic zeros and by defining a multiplicity framework. It establishes that quaternionic roots come in isolated points and spheres, counted with multiplicity to total $n$, and derives key identities such as $\prod_{m=1}^{n} |w_m| = \frac{|A_0|}{|A_n|}$ and $\sum_{m=1}^{n} {\rm Sc}(w_m) = -\frac{(A_n,A_{n-1})}{|A_n|^2}$, among others. A spherical-root criterion is provided via a recurrence-based function $Q_m$, $P_m$, namely $\sum_{m=1}^{n} A_m Q_m(w_0)=0$, and the paper shows the basic polynomials for right and left quaternionic polynomials coincide. Collectively, these results deepen the algebraic understanding of quaternionic polynomials and offer structured tools for root counting and characterization, bridging quaternionic root geometry with coefficient-based identities.

Abstract

The paper presents analogues of some Vieta formulas for quaternionic polynomials of the form $R_n(w)=\sum\limits_{m=0}^{n}A_mx^m$. A criterion for the sphericity of the root (zero) of a polynomial $R_n(w)$ is established.