Real Ghosts of Complex Hadamard Products

For all integers n≥1 and k≥2, the Hadamard product v1★⋯★vk of k elements of Kn+1 (with K being the complex numbers or real numbers) is the element v∈Kn+1 which is the coordinate-wise product of v1,…,vk (introduced by Cueto, Morton, and Sturmfels for a model in Algebraic Statistics). This product induces a rational map h:Pn(K)k⤏Pn(K). When K=C, k=2 and Xi(C)⊂Pn(C), i=1,2 are irreducible, we prove four theorems for the case dimX2(C)=1, three of them with X2(C) as a line. We discuss the existence (non-existence) of a cancellation law for ★-products and use the symmetry group of the Hadamard product. In the second part, we work over R. Under mild assumptions, we prove that by knowing X1(R)★⋯★Xk(R), we know X1(C)★⋯★Xk(C). The opposite, i.e., taking and multiplying a set of complex entries that are invariant for the complex conjugation and then seeing what appears in the screen Pn(R), very often provides real ghosts, i.e., images that do not come from a point of X1(R)×⋯×Xk(R). We discuss a case in which we certify the existence of real ghosts as well as a few cases in which we certify the non-existence of these ghosts, and ask several open questions. We also provide a scenario in which ghosts are not a problem, where the Hadamard data are used to test whether the images cover the full screen.