A Hierarchical Algebraic Framework for Analyzing Polynomial Automorphisms and the Jacobian Conjecture

This paper develops a hierarchical algebraic framework for analyzing polynomial automorphisms and their relationship to the Jacobian Conjecture. The Jacobian Conjecture, which asserts that a polynomial map F : Cn → Cn with constant non-zero Jacobian determinant has a polynomial inverse, remains one of the most challenging open problems in algebraic geometry. Our approach provides a systematic methodology for analyzing the structure of polynomial systems arising from the Jacobian Conjecture, with particular focus on elimination ideals, field extensions, and Galois actions. Weestablish several new theoretical results connecting the constant Jacobian condition to structural properties of polynomial systems and provide a characterization of polynomial automorphisms within our framework. While not claiming a complete proof of the Jacobian Conjecture, we demonstrate that our hierarchical analysis offers valuable insights and algorithmic tools for studying this longstanding problem. We validate our approach through rigorous mathematical analysis and carefully constructed examples, including both automorphisms and potential non-automorphisms.