No abstract

Given a surface defined as the graph of a real polynomial in two variables, we analyze some basic subsets characterized by its tangential singularities. If the parabolic curve is compact we provide certain criteria to determine when the unbounded component of its complement is hyperbolic. Moreover, we obtain an upper bound of the number of Gaussian cusps that holds even if the parabolic curve is non compact.

In this paper we study the affine geometric structure of the graph of a polynomial f ∈ R[x, y]. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of both fields of asymptotic lines and the Poincaré index at its singular points at infinity. We exhibit an index formula for the field of asymptotic lines involving the number of connected components of the projective Hessian curve of f and the number of godrons. As an application of this investigation, we obtain upper bounds, respectively, for the number of godrons having an interior tangency and when they have an exterior tangency.

No abstract

We study some realization problems related to the Hessian polynomials. In particular, we solve the Hessian curve realization problem for degrees zero, one, two, and three and the Hessian polynomial realization problem for degrees zero, one, and two.