Imaginary Projections: Complex Versus Real Coefficients

Abstract

Given a complex multivariate polynomial p ∈ C[z 1 , . . . , z n ], the imaginary projection I(p) of p is defined as the projection of the variety V(p) onto its imaginary part. We give a full characterization of the imaginary projections of conic sections with complex coefficients, which generalizes a classification for the case of real conics. More precisely, given a bivariate complex polynomial p ∈ C[z 1 , z 2 ] of total degree two, we describe the number and the boundedness of the components in the complement of I(p) as well as their boundary curves and the spectrahedral structure of the components. We further study the imaginary projections of some families of higher degree complex polynomials. In particular, we show a realizability result for strictly convex complement components which is in sharp contrast to the situation for real polynomials.