On an Arnold’s Conjecture Concerning the Space of Hyperbolic Homogeneous Polynomials

The set of homogeneous polynomials of degree D is a topological space that contains the subset Hyp ( D ) constituted only by hyperbolic polynomials. In 2002, V. I. Arnold conjectured in [1] that the number of connected components of Hyp ( D ) increases, as D increases, at least as a linear function of D . In this paper we prove that this conjecture is true. We determine the exact number of connected components of Hyp ( D ) and we provide a representative for each component. The proof is constructive; our approach uses homotopy invariance of the index of a curve and properties of homogeneous polynomials. We also describe some geometrical properties of the hyperbolic polynomials that we provide.