On moduli and arguments of roots of complex trinomials

Jan Čermák, Lucie Fedorková and Jiří Jánský

Vol. 332 (2024), No. 1, 39–67

DOI: 10.2140/pjm.2024.332.39

Abstract

Root properties of a general complex trinomial have been explored in numerous papers. Two questions have attracted a significant attention: the relationships between the moduli of these roots and the trinomial’s entries, and the location of the roots in the complex plane. We consider several particular problems connected with these topics, and provide new insights into them. As two main results, we describe the set of all trinomials having a root with a given modulus, and derive explicit formula for calculations of the arguments of such roots. In this fashion, we obtain a comprehensive characterization of these roots. In addition, we develop a procedure enabling us to compute moduli and arguments of all roots of a general complex trinomial with arbitrary precision. This procedure is based on the derivation of a family of real transcendental equations for the roots’ moduli, and it is supported by the formula for their arguments. All our findings are compared with the existing results.

Keywords

complex trinomial, root, location, modulus, argument

Mathematical Subject Classification

Primary: 12D10, 30C15

Secondary: 65H04

Milestones

Received: 17 October 2023

Revised: 30 September 2024

Accepted: 25 October 2024

Published: 20 November 2024