  The Hessian Topology is a subject with interesting relations with some
classical problems of analysis and geometry. In this article we prove a
conjecture on this subject stated by V.I. Arnold concerning the number of
connected components of hyperbolic homogeneous polynomials of degree $n$. The
proof is constructive and provides models. Our approach uses index properties
at isolated singularities of hyperbolic quadratic differential forms and
combinatorial properties of recurrent functions.
