2015/03/30 by Tuyen Trung Truong, Truong, Tuyen Trung · 2 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #Complex Variables (math.CV) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1503.08733
openalex publication_date 2015/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ≤ 9 satisfy the Jacobian conjecture.