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

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by 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

References

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