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arXiv · 2503.12219

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

Vinicio A. Gómez-Gutiérrez·Adriana Ortiz-Rodríguez
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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.

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Vinicio A. Gómez-Gutiérrez, Adriana Ortiz-Rodríguez. 2025-03-15. On an Arnold's Conjecture Concerning the Space of Hyperbolic Homogeneous Polynomials. https://arxiv.org/abs/2503.12219

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