Abstract: The global qualitative behaviour of fields of principal directions for the graph of a real-valued polynomial function f on the plane is studied. We determine and analyse the projective extension of these fields and show that they are defined by an analytic quadratic form on the whole unit 2-sphere. We prove that every umbilic point at infinity of this extension has a Poincaré-Hopf index equal to 1/2, and the topological type of a Lemon when the degree of f is 2 and the topological type of a Monstar for higher degrees. As a consequence, we prove a Poincaré-Hopf type formula for the graph of f such that, if all umbilics are isolated, the sum of all indices of the principal directions at umbilic points depends only upon the number of real linear factors of the homogeneous part of highest degree of f . A similar analysis is carried out in the case of f being a homogeneous polynomial.