Zero-Sets of Quaternionic and Octonionic Analytic Functions with Central Coefficients
Zero-Sets of Quaternionic and Octonionic Analytic Functions with Central Coefficients
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DOI：
10.1112/blms/19.4.329
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发表时间：
1987-07
影响因子：
0.9
通讯作者：
B. Datta;Subhashis Nag
中科院分区：
文献类型：
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作者：
B. Datta;Subhashis Nag
证明了任意具有中心系数（即真实的系数）的四元数（或八元数）解析函数f的零点集是R4或R8中两个余维球面与某些纯真实的点的不交并.特别地，对于具有真实的系数的多项式，完整的根集可以从根在复平面中的布局中几何地表征。根集变成有限数目的余维2欧几里得球面与有限数目的真实的点的并集。我们还找到了任何四元数（或八元数）A的原像f-1。我们证明，这一令人惊讶的现象，完整的领域的解决方案集的一部分是非常明显的一个特殊的“真实的”的现象。例如，四元数或八元数的任何非实数四元数（分别为八元数）的N次方根恰好是N个不同的点。所有这一切使我们能够为球面的自映射做一些有趣的拓扑。
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 Nth 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.