Publication result detail

On moduli and arguments of roots of complex trinomials

ČERMÁK, J.; FEDORKOVÁ, L.; JÁNSKÝ, J.

Original Title

On moduli and arguments of roots of complex trinomials

English Title

On moduli and arguments of roots of complex trinomials

Type

WoS Article

Original 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.

English 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

Key words in English

Complex trinomial, root, location, modulus, argument

Authors

ČERMÁK, J.; FEDORKOVÁ, L.; JÁNSKÝ, J.

RIV year

2025

Released

20.11.2024

Publisher

Mathematical Sciences Publishers

Location

Los Angeles, , USA

ISBN

0030-8730

Periodical

PACIFIC JOURNAL OF MATHEMATICS

Volume

332

Number

1

State

United States of America

Pages from

39

Pages to

67

Pages count

29

URL

https://msp.org/pjm/2024/332-1/pjm-v332-n1-p03-p.pdf

Full text in the Digital Library

http://hdl.handle.net/11012/250860

BibTex

@article{BUT191359,
  author="Jan {Čermák} and Lucie {Fedorková} and Jiří {Jánský}",
  title="On moduli and arguments of roots of complex trinomials",
  journal="PACIFIC JOURNAL OF MATHEMATICS",
  year="2024",
  volume="332",
  number="1",
  pages="39--67",
  doi="10.2140/pjm.2024.332.39",
  issn="0030-8730",
  url="https://msp.org/pjm/2024/332-1/pjm-v332-n1-p03-p.pdf"
}

Documents

pjm-v332-n1-p03-s