Some approaches toward the Jacobian conjecture

Abstract

In this paper, we study a so-called Condition C1 and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. Main results obtained: - Stating new equivalent formulations of the Jacobian conjecture. - Formulating some generalisations of the Jacobian conjecture and giving both theoretical and experimental evidences to support them. - Showing Condition C1 holds for a generic matrix of any given rank, is an invariant for a certain group action, and Condition C2 is an invariant for cubic similarity matrices. - Giving one heuristic argument for the truth of the Jacobian Conjecture. - Giving an effective (time saving) method to check whether a given Druzkowski map satisfies the Jacobian conjecture, explaining theoretically and checking on many examples including those previously considered by other authors. - Proposing approaches toward resolving the Jacobian conjecture. Showing that a generic Druzkowski map satisfies the criteria of some of these approaches (see Theorem 1.12), and hence expecting to be able to check these approaches for a given Druzkowski map very quickly. -As an application, proposing a strategy to use cubic similarity to check that Druzkowski maps of dimension $\leq 9$ satisfy the Jacobian conjecture.