A Counterexample to the Global Injectivity of a Jacobian Mapping in \mathbb{C}^5 and Its Analytical Roots

Technical Summary: A Counterexample to the Global Injectivity of a Jacobian Mapping in $\mathbb{C}^5$

Problem Statement
This paper addresses the Jacobian Conjecture, which posits that any polynomial mapping $F: \mathbb{C}^n \to \mathbb{C}^n$ with a constant non-zero Jacobian determinant is globally invertible. While reduction theorems by Bass, Connell, Wright, and Dru˙zkowski have narrowed the scope of the conjecture to cubic-linear (Dru˙zkowski) mappings, this study investigates a specific polynomial structure in $\mathbb{C}^5$ to test the limits of global injectivity under unipotent Jacobian conditions.

Methodology
The author constructs a specific polynomial mapping $G: \mathbb{C}^5 \to \mathbb{C}^5$ defined by $G(x) = x - \nabla F(x)$, where $F$ is a homogeneous polynomial of degree 6. The polynomial $F$ is composed of four "tridiagonal harmonic blocks" involving real and imaginary components derived from the real part of $(u+iv)^6$.

The methodology proceeds in three stages:

1. Jacobian Analysis: The author demonstrates that the Hessian matrix $H_F(x)$ of the defining polynomial is strictly tridiagonal. By analyzing the algebraic structure of the coefficients, it is shown that the traces of all powers of $H_F(x)$ vanish identically ($\text{Tr}(H_F^k) = 0$ for $k=1,\dots,5$). This proves $H_F(x)$ is nilpotent everywhere, ensuring the Jacobian determinant of the mapping $G$ is globally constant and equal to 1 ($\det(I - H_F) \equiv 1$).
2. Injectivity Testing: Leveraging the fact that $F$ consists of even-degree monomials, the gradient $\nabla F$ is an odd function. The author reduces the problem of finding distinct points $a \neq b$ such that $G(a) = G(b)$ to finding non-trivial roots of the equation $\nabla F(a) = a$ (i.e., $G(a) = \vec{0}$).
3. Analytical Classification: The paper performs a comprehensive derivation of the zero-set $G(x) = \vec{0}$. By isolating variables and applying proportionality ansatzes ($x_i = k x_j$) to specific subsystems, the author solves the resulting algebraic equations to identify distinct families of solutions.

Key Contributions and Results
The primary contribution of the paper is the construction of a counterexample to the global injectivity of a unipotent polynomial mapping in $\mathbb{C}^5$.

• Explicit Counterexample: The author constructs two distinct sparse vectors, $a = (c, 0, 0, 0, 0)$ and $b = (-c, 0, 0, 0, 0)$, where $c = 6^{-1/4}$. It is rigorously proven that $G(a) = G(b) = \vec{0}$, thereby demonstrating that the mapping is not globally injective despite having a Jacobian determinant of 1 everywhere.
• Classification of Zero-Sets: The paper provides a detailed classification of the analytical roots of the gradient field, identifying a total of 37 precise complex solutions organized into six distinct geometric series:
  1. Series 1: Four isolated solutions where only the first coordinate is non-zero ($x_1^4 = 1/6$).
  2. Series 2: Four solutions where only the last coordinate is non-zero ($x_5^4 = i/6$).
  3. Series 3: Four solutions where only the central coordinate is non-zero ($x_3^4 = 1/6$).
  4. Series 4: Sixteen solutions formed by independent linear combinations of the first and third coordinates.
  5. Series 5: A family of solutions on the right boundary involving coupled variables $x_4$ and $x_5$, determined by a quadratic equation in the scaling coefficient $k_R$.
  6. Series 6: A symmetric family of solutions on the left boundary involving coupled variables $x_1$ and $x_2$, determined by a quadratic equation in the scaling coefficient $k_L$.

Significance
The paper claims that these results highlight the complex nature of stability and injectivity conditions within the framework of the Jacobian Conjecture. By exhibiting a unipotent mapping in $\mathbb{C}^5$ that fails to be globally injective, the work underscores the difficulty of the conjecture and the richness of the algebraic solution space even when the Jacobian determinant is trivially constant. The findings are presented as a verification of non-injectivity for this specific class of mappings, offering precise analytical families of complex roots that challenge the assumption that constant Jacobian determinants imply global invertibility in higher dimensions.