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 \cite{arn0} 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.