An application of a nonuniform global stability problem to the study of parametrized polynomial automorphisms

TL;DR

This paper explores the relationship between injectivity in parametrized maps and a global stability conjecture, demonstrating that certain polynomial automorphisms have polynomial inverses under specific conditions, akin to the Jacobian Conjecture.

Contribution

It introduces new definitions of injectivity for parametrized maps and links them to a stability conjecture, proving polynomial invertibility for a family of automorphisms.

Findings

Parametrized polynomial automorphisms have polynomial inverses for specific parameters.

New injectivity definitions are proposed and linked to stability conjectures.

The work relates to the Jacobian Conjecture in polynomial invertibility.

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.