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On the Theorem of Gauss--Lucas for quaternions

I. Emizh, A. Guterman

TL;DR

The paper addresses extending the Gauss–Lucas theorem to quaternionic polynomials by projecting onto complex subplanes. It introduces the snail/cosnail-like framework and proves that $R(P') \subset \mathfrak{snm}(P) = \bigcup_{I\in\mathcal{S}} (\mathbf{conv}(R(P_I)) \cap \mathbf{conv}(R(P_I^{\perp})))$, strengthening prior results by GRPA. It also derives a tighter bound for the root moduli: $\max_{z\in R(P')} |z| \le \sup_{I\in\mathcal{S}} \min(C(P_I), C(P_I^{\perp}))$, with explicit examples showing the improvement. The approach leverages the isomorphism $\mathbb{R}[I] \cong \mathbb{C}$ to apply classical complex Gauss–Lucas results on each plane and then aggregates the information over all imaginary directions, yielding sharper geometric and quantitative estimates.

Abstract

It is proved that the roots of the derivative of a polynomial with quaternionic coefficients belong to the union of the intersections of sets defined in terms of certain projections of a polynomial. The result strengthens the quaternion version of Gauss-Lucas theorem, proved by Ghiloni and Perotti in 2018.

On the Theorem of Gauss--Lucas for quaternions

TL;DR

The paper addresses extending the Gauss–Lucas theorem to quaternionic polynomials by projecting onto complex subplanes. It introduces the snail/cosnail-like framework and proves that , strengthening prior results by GRPA. It also derives a tighter bound for the root moduli: , with explicit examples showing the improvement. The approach leverages the isomorphism to apply classical complex Gauss–Lucas results on each plane and then aggregates the information over all imaginary directions, yielding sharper geometric and quantitative estimates.

Abstract

It is proved that the roots of the derivative of a polynomial with quaternionic coefficients belong to the union of the intersections of sets defined in terms of certain projections of a polynomial. The result strengthens the quaternion version of Gauss-Lucas theorem, proved by Ghiloni and Perotti in 2018.
Paper Structure (5 sections, 14 theorems, 59 equations, 1 figure)

This paper contains 5 sections, 14 theorems, 59 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary information about quaternions
  3. Preliminary information about polynomials
  4. The Gauss-Lucas theorem for quaternions
  5. Estimation of the norm of the roots

Key Result

Theorem 1.1

Let $f \in \mathbb{C}[x]\setminus \mathbb{C}$ be an arbitrary non-constant polynomial of one variable with complex coefficients. Then the roots of the derivative $f'$ of the polynomial $f$ belong to the convex hull of the roots of the original polynomial, see PV, here $\mathbb{C}$ denote the field o

Figures (1)

  • Figure 1: Set $\mathbf{conv} (R(P_{i}))$ is a triangle $ABC$. Set $\mathbf{conv} (R(P_{i}^{\perp}))$ is a segment $DE$.

Theorems & Definitions (26)

  • Theorem 1.1: Gauss -- Lucas
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Example 2.4
  • Remark 2.5
  • Definition 3.1
  • Example 3.2
  • ...and 16 more