Given a multivariate complex polynomial pβC[z1β,β¦,znβ],
the imaginary projection I(p) of p is defined as the projection
of the variety 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 pβC[z1β,z2β]
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 show a
realizability result for strictly convex complement components which is in
sharp contrast to the case of real polynomials.Comment: 24 pages; Revised versio