Locating trinomial zeros

Howell, Russell; Kyle, David

doi:10.2140/involve.2018.11.711

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Locating trinomial zeros

Russell Howell and David Kyle

Vol. 11 (2018), No. 4, 711–720

DOI: 10.2140/involve.2018.11.711

Abstract

We derive formulas for the number of interior roots (i.e., zeros with modulus less than 1) and exterior roots (i.e., zeros with modulus greater than 1) for trinomials of the form $z^{n} + z^{k} - 1$ , where $1 \leq k \leq n - 1$ . Combined with earlier work by Brilleslyper and Schaubroeck, who focus on unimodular roots (i.e., zeros that lie on the unit circle), we give a complete count of the location of zeros of these trinomials.

Keywords

trinomials, complex analysis, Diophantine equations, zeros of functions

Mathematical Subject Classification 2010

Primary: 30C15, 97I80

Secondary: 11-02

Milestones

Received: 21 July 2017

Accepted: 29 August 2017

Published: 15 January 2018

Communicated by Michael Dorff

Authors

Russell Howell
Department of Mathematics Westmont College Santa Barbara, CA United States

David Kyle
Department of Mathematics Westmont College Santa Barbara, CA United States

Vol. 11, No. 4, 2018