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

We study a specific polynomial mapping G: \mathbb{C}^5 \to \mathbb{C}^5 induced by a homogeneous polynomial F of degree 6 consisting of four tridiagonal harmonic blocks.

We prove that while the Jacobian matrix of this mapping is unipotent at every point (implying det J_G(x) \equiv 1), the mapping itself is not globally injective.

We construct explicit algebraic sparse pairs of distinct points a \neq b that map to the identical image G(a) = G(b) = \vec{0}.

Furthermore, we perform a comprehensive classification of the zero-set structure of the corresponding gradient field, uncovering a total of 37 precise analytical complex solutions split across six distinct geometric series.