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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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An Algorithm for the Factorization of Split Quaternion Polynomials.

Daniel F Scharler1, Hans-Peter Schröcker1

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This study introduces an algorithm for factoring univariate polynomials over split quaternions, offering geometric insights into non-factorizability and enabling decomposition of rational motions into hyperbolic rotations.

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Area of Science:

  • Algebraic Geometry
  • Quaternion Theory
  • Geometric Kinematics

Background:

  • Split quaternions are algebraic structures with applications in describing rational motions in hyperbolic geometry.
  • The factorization of polynomials over non-commutative rings presents unique challenges and theoretical implications.
  • Understanding polynomial factorization is crucial for decomposing complex motions into simpler components.

Purpose of the Study:

  • To develop an algorithm for computing all linear factorizations of univariate polynomials over split quaternions.
  • To provide geometric interpretations for polynomials that cannot be factored over split quaternions.
  • To extend factorization techniques to dual quaternion polynomials describing motions in Euclidean kinematics.

Main Methods:

  • Development of a novel algorithm for polynomial factorization over split quaternions.
  • Geometric analysis of non-factorizable polynomials using rulings on a quadric surface.
  • Investigation of real polynomial multiples for factorizable split quaternion polynomials.
  • Adaptation of split quaternion factorization techniques for dual quaternion polynomials.

Main Results:

  • An algorithm is presented to compute all linear factorizations of univariate polynomials over split quaternions.
  • Geometric interpretations for non-factorizability are established via rulings on the quadric of non-invertible split quaternions.
  • Methods are described to find real polynomial multiples that allow factorization, preserving the described motion.
  • New factorizations for certain dual quaternion polynomials are computed by transferring techniques from split quaternions.

Conclusions:

  • The algorithm successfully computes factorizations, linking polynomial structure to the decomposition of rational motions into hyperbolic rotations.
  • The geometric interpretations offer insights into the limitations and properties of split quaternion polynomials.
  • The extension to dual quaternions broadens the applicability of these factorization methods to Euclidean kinematics.