Properties of Trinomials of Height at least 2

TL;DR

This paper investigates the algebraic and number-theoretic properties of certain trinomials, focusing on their irreducibility, Mahler measure, and house, especially when the height is at least 2.

Contribution

It provides new insights into the irreducibility and Mahler measure of trinomials with specific coefficient constraints, extending understanding of their algebraic properties.

Findings

Conditions for irreducibility of the trinomials.

Bounds on Mahler measure for these polynomials.

Characterization of the house of the trinomials.

Abstract

This paper is concerned with trinomials of the form $z^{n}+az^{m}+b$, where $0<m<n$ are relatively prime integers and $a$ and $b$ are non-zero complex numbers. Typically (but not exclusively), $a$ will be an integer with $|a| \geq 2$, while $b=\pm 1$. Our main results cover the irreducibility, Mahler measure and house of such trinomials.