Imaginary projections of polynomials

TL;DR

This paper introduces the imaginary projection of polynomials as a geometric tool to analyze polynomial stability, revealing convexity properties and classifying quadratic cases, thus connecting algebraic geometry with stability analysis.

Contribution

It defines the imaginary projection of polynomials, explores its geometric properties, and provides a complete classification for quadratic polynomials, advancing the understanding of polynomial stability.

Findings

Connected components of the complement of imaginary projections are convex.

Complete classification of imaginary projections for quadratic polynomials.

Characterization of limit directions for polynomials of arbitrary degree.

Abstract

We introduce the imaginary projection of a multivariate polynomial $f \in \mathbb{C}[\mathbf{z}]$ as the projection of the variety of $f$ onto its imaginary part, $\mathcal{I}(f) \ = \ \{\text{Im}(\mathbf{z}) \, : \, \mathbf{z} \in \mathcal{V}(f) \}$. Since a polynomial $f$ is stable if and only if $\mathcal{I}(f) \cap \mathbb{R}_{>0}^n \ = \ \emptyset$, the notion offers a novel geometric view underlying stability questions of polynomials. We show that the connected components of the complement of the closure of the imaginary projections are convex, thus opening a central connection to the theory of amoebas and coamoebas. Building upon this, the paper establishes structural properties of the components of the complement, such as lower bounds on their maximal number, proves a complete classification of the imaginary projections of quadratic polynomials and characterizes the limit directions for polynomials of arbitrary degree.