On the Arnold's conjecture on hyperbolic homogeneous polynomials

TL;DR

This paper proves Arnold's conjecture regarding the number of connected components of hyperbolic homogeneous polynomials of degree n, using a constructive approach involving index theory and combinatorics.

Contribution

It provides a constructive proof of Arnold's conjecture on hyperbolic homogeneous polynomials, including explicit models and a novel approach using index and combinatorial methods.

Findings

Confirmed the conjecture on the number of connected components

Developed models illustrating the conjecture's validity

Introduced a new approach combining index theory and combinatorics

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.