2025/03/15 by Gómez-Gutiérrez, Vinicio A., Ortiz-Rodríguez, Adriana
#26C05 #53A05 #54C30 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.12219
The set of homogeneous polynomials of degree D is a topological space that contains the subspace Hyp(D) constituted only by hyperbolic polynomials. In 2002, V. I. Arnold conjectured 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.