Imaginary Projections: Complex Versus Real Coefficients

TL;DR

This paper investigates the imaginary projections of complex polynomials, providing explicit characterizations for certain families, especially conic sections, and revealing unique convexity properties not seen in real polynomials.

Contribution

It offers a comprehensive classification of the imaginary projections of complex conics, extending real conic results and exploring their geometric and spectral properties.

Findings

Full characterization of imaginary projections for complex conic sections.

Description of the number and boundedness of components in the complement of the projections.

Demonstration of realizability of strictly convex complement components.

Abstract

Given a multivariate complex polynomial ${p\in\mathbb{C}[z_1,\ldots,z_n]}$, the imaginary projection $\mathcal{I}(p)$ of $p$ is defined as the projection of the variety $\mathcal{V}(p)$ onto its imaginary part. We focus on studying the imaginary projection of complex polynomials and we state explicit results for certain families of them with arbitrarily large degree or dimension. Then, we restrict to complex conic sections and give a full characterization of their imaginary projections, which generalizes a classification for the case of real conics. That is, given a bivariate complex polynomial $p\in\mathbb{C}[z_1,z_2]$ of total degree two, we describe the number and the boundedness of the components in the complement of $\mathcal{I}(p)$ as well as their boundary curves and the spectrahedral structure of the components. We further show a realizability result for strictly convex complement components which is in sharp contrast to the case of real polynomials.