On the Arnold's conjecture on hyperbolic homogeneous polynomials

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.