On the geometric structure of certain real algebraic surfaces

In this paper we study the affine geometric structure of the graph of a polynomial $f \in \mathbb{R} [x,y]$. 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 Poincar\'e index of its singular points when the surface is generic. Thus, 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 the special parabolic points. As an application of this investigation, we obtain upper bounds, respectively, for the number of special parabolic points having an interior tangency and when they have an exterior tangency.