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Zero‐Sets of Quaternionic and Octonionic Analytic Functions with Central Coefficients
Author(s) -
Datta Basudeb,
Nag Subhashis
Publication year - 1987
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/19.4.329
Subject(s) - mathematics , quaternion , codimension , spheres , zero (linguistics) , pure mathematics , disjoint sets , plane (geometry) , euclidean geometry , mathematical analysis , set (abstract data type) , geometry , linguistics , philosophy , physics , astronomy , computer science , programming language
We prove that the zero set of any quaternionic (or octonionic) analytic function f with central (that is, real) coefficients is the disjoint union of codimension two spheres in R 4 or R 8 (respectively) and certain purely real points. In particular, for polynomials with real coefficients, the complete root‐set is geometrically characterisable from the lay‐out of the roots in the complex plane. The root‐set becomes the union of a finite number of codimension 2 Euclidean spheres together with a finite number of real points. We also find the preimages f −1 for any quaternion (or octonion) A . We demonstrate that this surprising phenomenon of complete spheres being part of the solution set is very markedly a special ‘real’ phenomenon. For example, the quaternionic or octonionic N th roots of any non‐real quaternion (respectively octonion) turn out to be precisely N distinct points. All this allows us to do some interesting topology for self‐maps of spheres.