An application of a nonuniform global stability problem to the study of parametrized polynomial automorphisms
Álvaro Castañeda, Ignacio Huerta, Gonzalo Robledo
TL;DR
The paper links a nonuniform stability framework for nonautonomous systems to injectivity properties of parametrized map families, proposing B-NUMYC as a bridge to the Jacobian Conjecture. It introduces four injectivity notions, proves that B-NUMYC implies partial injectivity, and applies this to polynomial automorphisms arising from f(t, x) = x + H(t, x) with cubic homogeneous, nilpotent nonlinearities. A central contribution is Theorem FT, which, under B-NUMYC and small nonlinear perturbations, yields partial injectivity and polynomial inverses for fixed parameter values, connecting JC to a nonautonomous stability context via reduction arguments. The work further provides a concrete illustrative example where partial injectivity and a polynomial inverse are explicitly obtained, showcasing a nonautonomous route to JC and highlighting the role of the nonuniform dichotomy spectrum in guiding stability-based injectivity results.
Abstract
We propose a handful of definitions of injectivity for a parametrized family of maps and study its link with a global nonuniform stability conjecture for nonautonomous differential systems, which has been recently introduced. This relation allow us to address a particular family of parametrized polynomial automorphisms and to prove that they have polynomial inverse for certain parameters, which is reminiscent to the Jacobian Conjecture.