The Jacobian Conjecture and Integrability of Associated Partial Differential Equations

TL;DR

This paper links the Jacobian conjecture to integrability of associated PDEs, deriving broad polynomial map families satisfying the conjecture and proposing a systematic approach to solve the problem through PDE integrability.

Contribution

It introduces a novel PDE-based framework for the Jacobian conjecture, enabling systematic integration and broad polynomial solutions in all dimensions.

Findings

Derived broad families of polynomial maps satisfying the conjecture

Reformulated the conjecture into a system of integrable subequations

Proposed a systematic method for solving the parametrized Jacobian problem

Abstract

The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work obtains broad families of polynomial maps satisfying the conjecture in all dimensions and of arbitrarily high degrees. Furthermore, it is shown that a reformulated multiply parametrized version of the conjecture in all dimensions enables a separation of the Jacobian equation into a system of subequations which may be integrated systematically rendering a settlement of the parametrized Jacobian problem in this context.