The weak Markus--Yamabe conjecture fails in dimension 14

Technical Summary: The Weak Markus–Yamabe Conjecture Fails in Dimension 14

Problem Statement
The paper addresses the Weak Markus–Yamabe Conjecture, which posits that any $C^1$ map $X: \mathbb{R}^n \to \mathbb{R}^n$ is injective if its Jacobian matrix $JX(x)$ has eigenvalues with strictly negative real parts for all $x \in \mathbb{R}^n$ (a "Hurwitz map"). While the classical conjecture (asserting the singularity is a global attractor) is known to be false for $n > 2$, the weak version regarding injectivity remained open for polynomial maps in dimensions $n \ge 3$. The authors aim to determine if the weak conjecture holds for polynomial Hurwitz maps, specifically investigating the connection between this conjecture and the Jacobian Conjecture.

Methodology
The authors employ two distinct algebraic constructions to transform non-injective polynomial maps (Keller maps) into polynomial vector fields with Hurwitz Jacobians that possess multiple singularities.

1. Chain Realization Theorem (Theorem 1.2):

   • The authors introduce a "chain realization" mechanism that converts a polynomial Keller map $F = \text{Id} + H$ (where $H$ has no constant or linear terms) into a vector field $X_{F, \beta}$ on a higher-dimensional space $\mathbb{R}^N$.
   • The dimension is calculated as $N = \sum_{i=1}^n \max(d_i, 2) - n$, where $d_i$ are the degrees of the components of $F$.
   • The construction utilizes a graded similarity lemma (Lemma 2.6), a generalization of the scaling arguments used for cubic homogeneous maps. This lemma proves that if the determinant of $I - M(u)$ is constant, then $M(u)$ is nilpotent.
   • By applying this to a Keller map with a non-trivial fiber (multiple points mapping to the same image), the resulting vector field $X$ has a Jacobian spectrum of $\{-1\}$ everywhere (making it Hurwitz) and a singular set in polynomial bijection with the fiber of $F$.

2. Rank-Reduced Dehomogenization (Section 7):

   • A second construction minimizes the degree of the resulting vector field rather than the dimension.
   • Starting with a cubic stabilization of a Keller map, the authors apply a rank-reduction technique to the cubic homogeneous part. By decomposing the cubic term $C$ into $E\tilde{C}$ based on the component rank, they construct a map $K_E$ on an extended space.
   • This method produces a Hurwitz vector field of degree three, independent of the chain construction.

Key Contributions and Results

• Counterexample in Dimension 14:
  Applying the Chain Realization Theorem to the recent explicit counterexample to the Jacobian Conjecture announced by Alpöge (a non-injective Keller map $F: \mathbb{R}^3 \to \mathbb{R}^3$ with rational coefficients and three distinct points in the same fiber), the authors construct an explicit polynomial vector field $X: \mathbb{R}^{14} \to \mathbb{R}^{14}$.

  • Properties: The field is of degree seven. Its Jacobian matrix has a spectrum of $\{-1\}$ at every point.
  • Singularities: The field possesses three distinct rational singularities ($S_1, S_2, S_3$).
  • Conclusion: Since a Hurwitz map with multiple singularities cannot be injective, the Weak Markus–Yamabe Conjecture is false in dimension 14. By extending the vector field trivially, the conjecture is shown to fail for all dimensions $n \ge 14$.

• Counterexample in Dimension 18 (Degree 3):
  Using the rank-reduced dehomogenization on the same underlying map, the authors construct a polynomial Hurwitz vector field on $\mathbb{R}^{18}$.

  • Properties: This field is of degree three and also possesses three distinct rational singularities.

• Discrete Counterexample:
  The paper notes that the map $G = \text{Id} + X$ (where $X$ is the dimension 14 field) has a nilpotent Jacobian at every point and three locally asymptotically stable fixed points, disproving the discrete analog of the problem.

Significance and Claims
The paper establishes that the Weak Markus–Yamabe Conjecture fails within the class of polynomial maps for dimensions $n \ge 14$. The significance lies in the explicit construction of these counterexamples using the recent breakthrough on the Jacobian Conjecture.

• Reversal of Logic: Historically, the connection between the Jacobian Conjecture and the Weak Markus–Yamabe Conjecture was viewed as: "If the Weak Conjecture is true, then the Jacobian Conjecture is true." This paper demonstrates the converse mechanism: "Since the Jacobian Conjecture is false (via Alpöge's counterexample), the Weak Conjecture must also be false in sufficiently high dimensions."
• Nature of the Obstruction: Unlike classical counterexamples to the strong Markus–Yamabe conjecture (which involve orbits escaping to infinity), the obstruction here is the coexistence of multiple equilibria (hyperbolic sinks). The vector fields are locally contracting everywhere but fail to be globally injective due to the presence of multiple basins of attraction.
• Open Dimensions: The authors explicitly state that their constructions do not resolve the status of the conjecture for dimensions $3 \le n \le 13$. The dimension 14 result depends on the specific degree structure of the Alpöge map; minimizing the dimension further would require finding a Keller map with lower component degrees or optimizing the map via polynomial automorphisms, which remains an open problem.

The paper concludes that while the algebraic collision of fibers in the Jacobian Conjecture context translates directly into a polynomial dynamical system with multiple hyperbolic sinks, the specific dimensional threshold for the lowest possible counterexample remains undetermined.