An introduction to upper half plane polynomials

TL;DR

This paper introduces the properties of multivariable polynomials that do not vanish in the open upper half plane, generalizing real-rooted polynomial properties and providing foundational insights into their behavior.

Contribution

It offers a straightforward introduction to upper half plane polynomials, expanding understanding of their properties and generalizations from real-rooted polynomials.

Findings

Characterization of upper half plane polynomials

Connections to real-rooted polynomial properties

Foundational insights into multivariable polynomial behavior

Abstract

This is a straightforward introduction to the properties of polynomials in many variables that do not vanish in the open upper half plane. Such polynomials generalize many of the well-known properties of polynomials with all real roots.