Abstract
The roots of polynomials over Cayley–Dickson algebras over an arbitrary field and of arbitrary dimension are studied. It is shown that the spherical roots of a polynomial f(x) are also roots of its companion polynomial (Formula presented.). We generalize the classical theorems for complex and real polynomials by Gauss–Lucas and Jensen to locally-complex Cayley–Dickson algebras: it is proved that the spherical roots of (Formula presented.) belong to the convex hull of the roots of (Formula presented.), and we also show that all roots of (Formula presented.) are contained in the snail of f(x), as defined by Ghiloni and Perotti.
| Original language | English |
|---|---|
| Pages (from-to) | 1355-1369 |
| Number of pages | 15 |
| Journal | Communications in Algebra |
| Volume | 51 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2023 |
Bibliographical note
Publisher Copyright:© 2022 Taylor & Francis Group, LLC.
Funding
The first two authors thank the organizers of the CIMPA 2020 school on Non-associative Algebras and their Applications (taking place in August 2021), which triggered this collaboration. The work of the fourth author was partially financially supported by the grant RSF 21-11-00283. The authors thank the anonymous referee for the helpful comments.
Keywords
- Cayley–Dickson algebras
- Gauss–Lucas theorem
- Jensen’s theorem
- locally-complex algebras
- octonion algebras