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

Abstract

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.