[Paper Review] Some approaches toward the Jacobian conjecture

Tuyen Trung Truong|arXiv (Cornell University)|Mar 30, 2015

Advanced Differential Equations and Dynamical Systems9 references3 citations

TL;DR

This paper proposes Conditions C1 and C2 as computationally tractable criteria to verify the Jacobian Conjecture for Druzkowski maps—polynomial maps of the form $F(x) = x + (l_i(x))^3$ with linear forms $l_i$. It shows that satisfying Condition C2 implies the Jacobian Conjecture for such maps, provides an efficient algorithm using Gröbner bases in Mathematica, and proves that generic Druzkowski matrices satisfy these conditions, offering a practical pathway to verify the conjecture up to dimension 9.

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.

Motivation & Objective

• To develop computationally feasible criteria to verify the Jacobian Conjecture for Druzkowski maps, which are central to the generalized Jacobian Conjecture.
• To address the lack of systematic methods for checking the conjecture in higher dimensions, especially beyond dimension 8.
• To provide theoretical and experimental evidence that Conditions C1 and C2 are effective in verifying injectivity and polynomial invertibility of Druzkowski maps.
• To offer a practical computational framework using Gröbner bases and Mathematica codes to test these conditions on specific matrices.
• To propose a strategy using cubic similarity to reduce the verification of the Jacobian Conjecture for Druzkowski maps of dimension ≤9 to simpler cases.

Proposed method

• Introduces Condition C1 and Condition C2 as equivalent formulations of the Jacobian Conjecture for Druzkowski maps, with C2 being a weaker but sufficient condition.
• Uses the structure of Druzkowski matrices—defined by $A$ with $A^2 = 0$—to derive polynomial equations encoding the injectivity and invertibility of the map $F(x) = x + (l_i(x))^3$.
• Employs Gröbner basis computation over polynomial ideals generated by the components of $F(x) - F(y)$ and the determinant condition $\det(\mathrm{Id} + \Delta((sx+ty)^2) \cdot A) = 1$ to test for triviality of the solution set.
• Applies a heuristic argument based on genericity: a generic Druzkowski matrix of any rank satisfies Condition C1, suggesting that verification can be fast in practice.
• Implements a Mathematica-based algorithm that checks whether a given Druzkowski matrix satisfies Condition C2 by computing Gröbner bases of the associated polynomial system.
• Utilizes cubic similarity transformations to reduce the verification problem: if a matrix is cubically similar to one satisfying known criteria, it inherits the Jacobian Conjecture property.

Experimental results

Research questions

• RQ1 Can Conditions C1 and C2 serve as effective, computationally efficient criteria to verify the Jacobian Conjecture for Druzkowski maps?
• RQ2 Does the validity of Condition C2 for a Druzkowski matrix imply the existence of a polynomial inverse, thus confirming the Jacobian Conjecture for that map?
• RQ3 Can the proposed method using Gröbner bases and symbolic computation in Mathematica be systematically applied to verify the conjecture for Druzkowski maps of dimension up to 9?
• RQ4 Is the property of satisfying Condition C2 invariant under cubic similarity transformations, enabling reduction to simpler cases?
• RQ5 Does the genericity of Druzkowski matrices in satisfying Condition C1 suggest that most such matrices can be quickly verified as satisfying the Jacobian Conjecture?

Key findings

• Condition C2 is equivalent to the Jacobian Conjecture for Druzkowski maps, providing a practical criterion for verification.
• A generic Druzkowski matrix of any given rank satisfies Condition C1, indicating that most such matrices are likely to satisfy the conjecture.
• The proposed method using Gröbner bases in Mathematica significantly reduces computation time for checking injectivity and the determinant condition, as demonstrated on multiple test cases.
• The author proves that Condition C2 is invariant under cubic similarity, enabling a strategy to verify the conjecture for all Druzkowski maps of dimension ≤9 by checking only a representative set.
• Theoretical and computational evidence supports that the Jacobian Conjecture holds for all Druzkowski maps of dimension ≤9, using the proposed approach based on cubic similarity and Condition C2.
• The paper provides explicit Mathematica code to check whether a given Druzkowski matrix satisfies Condition C2, with timing results showing efficiency even for higher-dimensional cases.