On the number of roots for harmonic trinomials

In this manuscript we study the counting problem for harmonic trinomials of the form \(a\zeta^n+b\overline{\zeta}^m+c\), where \(n,m\in \mathbb{N}\), \(n>m\), and \(a\), \(b\) and \(c\) are non-zero complex numbers. As a consequence, we obtain the Fundamental Theorem of Algebra and the Wilmshurst...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Barrera, Gerardo, Barrera, Waldemar, Juan Pablo Navarrete
Format: Article
Language:English
Subjects:
Complex numbers
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Summary:In this manuscript we study the counting problem for harmonic trinomials of the form \(a\zeta^n+b\overline{\zeta}^m+c\), where \(n,m\in \mathbb{N}\), \(n>m\), and \(a\), \(b\) and \(c\) are non-zero complex numbers. As a consequence, we obtain the Fundamental Theorem of Algebra and the Wilmshurst conjecture for harmonic trinomials. The proof of the counting problem relies on the Bohl method introduced in Bohl (1908).
ISSN:2331-8422
DOI:10.48550/arxiv.2112.06703