On the Complex and Real Hessian Polynomials

A topic which has been of interest since the XIX century is the study of the parabolic curve of smooth surfaces in real three-dimensional space, as shown in the works of Gauss, Darboux, Salmon 1 , Kergosien and Thom 2 , Arnold 3 , among others. The parabolic curve of the graph of a smooth function, f : R2 → R, is the set { p, f p ∈ R2 × R: Hessf p 0}, where Hess f : fxxfyy − f2 xy. In this case, the Hessian curve of f , Hess f x, y 0, is a plane curve which is the projection of the parabolic curve into the xy-plane along the z-axis. When f is a polynomial of degree n in two variables, Hess f is a polynomial of degree at most 2n − 4. Therefore, the Hessian curve is an algebraic plane curve. In this setting there are two natural realization problems related to the Hessian of a polynomial.