On the Arnold's conjecture on hyperbolic homogeneous polynomials

2013/01/11 by Adriana Ortiz-Rodríguez, Ortiz-Rodríguez, Adriana, Federico Sánchez-Bringas +1
Mathematics · #53A05 #53A15 #53E05 #Advanced Differential Equations and Dynamical Systems #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1301.2364

openalex publication_date 2013/01/11 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of hyperbolic homogeneous polynomials of degree n. The proof is constructive and provides models. Our approach uses index properties at isolated singularities of hyperbolic quadratic differential forms and combinatorial properties of recurrent functions.

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