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Quadratic unilateral polynomials over split quaternions

Wensheng Cao

Abstract

In this paper, we derive explicit formulas for computing the roots of $ax^{2}+bx+c=0$ with $a$ being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results when the corresponding companion polynomials are not identically vanishing.

Quadratic unilateral polynomials over split quaternions

Abstract

In this paper, we derive explicit formulas for computing the roots of with being not invertible in split quaternion algebra. We also imitate the approach developed by Opfer, Janovska and Falcao etc. to verify our results when the corresponding companion polynomials are not identically vanishing.
Paper Structure (14 sections, 19 theorems, 211 equations, 2 tables)

This paper contains 14 sections, 19 theorems, 211 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Split quaternions
  3. Niven's algorithm over ${\mathbb H}_s$
  4. Quadratic equation in ${\mathbb H}_s$
  5. Equation I
  6. Equation II for SZ
  7. Case $P_{ab}\neq 0$
  8. Case $P_{ab}=0$
  9. Subcase $P_{{\bf i} a ,b}=0$
  10. Subcase $P_{ a {\bf i},b}=0$
  11. Equation II for SI
  12. Case $P_{ab}\neq 0$
  13. Case $P_{ab}=0$
  14. Verification of our examples by companion polynomial approach

Key Result

Proposition 1.1

(c.f. cao) Two split quaternions $a,b \in {\mathbb H}_s-{\mathbb R}$ are similar if and only if

Theorems & Definitions (45)

  • Proposition 1.1
  • Proposition 1.2
  • Proposition 1.3
  • Example 1.1
  • Proposition 1.4
  • proof
  • Lemma 2.1
  • Proposition 2.1
  • proof
  • Theorem 2.1
  • ...and 35 more