Resultado de injetividade para aplicações diferenciáveis semialgébricas no R^2

Abstract

The Chamberland conjecture, presented in the article “A Mountain Pass to the Jacobian Conjecture” in 1998, states that if F : Rn → Rn is a C1 mapping such that for some ε > 0, the set of complex eigenvalues of its derivative does not intersect with the set {z ∈ C ; |z| < ε} , then F is injective. In this work, we address the article “Global Asymptotic Stability for Differentiable Vector Fields of R2 ”, by authors Alexandre Fernandes, Carlos Gutierrez, and Roland Rabanal in 2004, which proves the Chamberland conjecture for differentiable mappings in R2 . More specifically, we provide a reinterpretation of the proof of the Injectivity Result presented in this article, which states that given a differentiable mapping F in R2 and ε > 0, if the set of complex eigenvalues of its derivative does not intersect with the interval [0, ε), then it follows that F is injective. Additionally, we introduce the hypothesis that the mapping is semialgebraic, which allows us to derive significant implications from the original exposition, utilizing certain techniques made possible by semialgebraic sets and functions.