The Jacobian Conjecture and Integrability of Associated Partial Differential Equations

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.

27 pages. This article is an invited contribution for a memorial issue of Physica D in honor of Vladimir Zakharov