Imaginary projections: Complex versus real coefficients

Given a multivariate complex polynomial p∈C[z1,…,zn], the imaginary projection I(p) of p is defined as variety V(p) onto its part. We focus on studying polynomials and we state explicit results for certain families them with arbitrarily large degree or dimension. Then, restrict to conic sections give full characterization their projections, which generalizes classification case real conics. That is, given bivariate p∈C[z1,z2] total two, describe number boundedness components in complement well boundary curves spectrahedral structure components. further show realizability result strictly convex sharp contrast polynomials.