The paper presents a counterexample to the Jacobian Conjecture, a long-standing problem in algebraic geometry which posits that any polynomial map $F: \mathbb{C}^n \to \mathbb{C}^n$ with a non-zero constant Jacobian determinant must have a polynomial inverse. The author provides an explicit polynomial map $F: \mathbb{C}^3 \to \mathbb{C}^3$ defined by the mapping:

$F(x, y, z) = \left( (1+xy)^3 z + y^2 (1+xy) (4+3xy), \,\, y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), \,\, 2 x - 3 x^2 y - x^3 z \right)$

The core methodology involves constructing a map where the Jacobian determinant, $\det(JF)$, is calculated to be the constant value $-2$. Since the Jacobian determinant is a non-zero constant, the map satisfies the initial condition required by the conjecture.

To demonstrate that the map does not possess a polynomial inverse, the author identifies specific points in the domain that map to a single target point in the codomain. Specifically, the points $(0, 0, -1/4)$, $(1, -3/2, 13/2)$, and $(-1, 3/2, 13/2)$ are all mapped by $F$ to the image point $(-1/4, 0, 0)$. The existence of these distinct pre-images for a single point demonstrates that the map is not injective. Because a polynomial map with a polynomial inverse must be an automorphism and therefore injective, the failure of injectivity in this mapping provides a constructive disproof of the Jacobian Conjecture. The author credits Akhil and Fable for their contributions to the derivation and verification of this counterexample.