Roots of Elliptic Scator Numbers

Manuel Fernandez-Guasti

doi:10.3390/axioms10040321

Abstract

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

Highlights

We focus on the subset of scators with a zero additive scalar and director components, that is, null scators
A null or zero scator in terms of the multiplicative variables is obtained if (i) ζ 0 = 0 or (ii) in the additive representation, if two or more director arguments of a scator are equal to π2, mod π ([6], Section 2.4)
The generalization of the De Moivre formula to hypercomplex scator space permits the evaluation of the exponential of a scator in S1+2 ([6], Lemma 4)

Summary

Introduction

None of the q × n products should give a scator with a zero additive scalar component if two or more director coefficients are different from zero; oq ζ r1 r2 ···rn satisfies ζ r1 r2 ···rn = φ. A null or zero scator in terms of the multiplicative variables is obtained if (i) ζ 0 = 0 or (ii) in the additive representation, if two or more director arguments of a scator are equal to π2 , mod π ([6], Section 2.4). A scator of the form ζ = ζ 0 ∏nj6=l,q cos ζ j + sin ζ j ě j ěl ěq cannot be a o 2m root of a non-zero scator because ζ cannot be written as a product of 2m equal factors.

Square Roots in the Scator Set

Cube Roots of 1 o

Fourth Roots of 1 o

Fifth Roots of 1

Conclusions