# Niven’s Algorithm Applied to the Roots of the Companion Polynomial Over $${{\mathbb {R}}}^4$$ R 4 Algebras

Niven’s Algorithm Applied to the Roots of the Companion Polynomial Over $${{\mathbb {R}}}^4$$... This paper will contain an extension of Niven’s algorithm of 1941, which in its original form is designed for finding zeros of unilateral polynomials p over quaternions $${\mathbb {H}}$$ H . The extensions will cover the algebra $${{\mathbb {H}}_{\mathrm{coq}}}$$ H coq of coquaternions, the algebra $${{\mathbb {H}}_{\mathrm{nec}}}$$ H nec of nectarines and the algebra $${{\mathbb {H}}_{\mathrm{con}}}$$ H con of conectarines. These are nondivision algebras in $${{\mathbb {R}}}^4$$ R 4 . In addition, it is also shown that in all algebras the most difficult part of Niven’s algorithm can easily be solved by inserting the roots of the companion polynomial c of p, with the result, that all zeros of all unilateral polynomials over all noncommutative $${{\mathbb {R}}}^4$$ R 4 algebras can be found. In addition, for all four algebras the maximal number of zeros can be given. For the three nondivision algebras besides the known types of zeros: isolated, spherical, hyperbolic, a new type of zero will appear, which will be called unexpected zero of p. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

# Niven’s Algorithm Applied to the Roots of the Companion Polynomial Over $${{\mathbb {R}}}^4$$ R 4 Algebras

, Volume 27 (3) – Jun 13, 2017
17 pages

/lp/springer_journal/niven-s-algorithm-applied-to-the-roots-of-the-companion-polynomial-0JKi0J6F9u
Publisher
Springer International Publishing
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-017-0786-y
Publisher site
See Article on Publisher Site

### Abstract

This paper will contain an extension of Niven’s algorithm of 1941, which in its original form is designed for finding zeros of unilateral polynomials p over quaternions $${\mathbb {H}}$$ H . The extensions will cover the algebra $${{\mathbb {H}}_{\mathrm{coq}}}$$ H coq of coquaternions, the algebra $${{\mathbb {H}}_{\mathrm{nec}}}$$ H nec of nectarines and the algebra $${{\mathbb {H}}_{\mathrm{con}}}$$ H con of conectarines. These are nondivision algebras in $${{\mathbb {R}}}^4$$ R 4 . In addition, it is also shown that in all algebras the most difficult part of Niven’s algorithm can easily be solved by inserting the roots of the companion polynomial c of p, with the result, that all zeros of all unilateral polynomials over all noncommutative $${{\mathbb {R}}}^4$$ R 4 algebras can be found. In addition, for all four algebras the maximal number of zeros can be given. For the three nondivision algebras besides the known types of zeros: isolated, spherical, hyperbolic, a new type of zero will appear, which will be called unexpected zero of p.

### Journal

Advances in Applied Clifford AlgebrasSpringer Journals

Published: Jun 13, 2017

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