Preprint provides uniform construction of polynomial maps with non-injective characteristics in dimension three, indicating the conjecture's limitations.
Key Points
This work aims to demonstrate that non-injective polynomial maps with constant nonzero Jacobians exist in all generic fiber degrees n ≥ 3 in dimension three.
Construction of a polynomial map Fₙ for every integer n ≥ 3 emphasizing a uniform one-variable framework.
Explicit calculation of the Jacobian determinant, showing it is a nonzero constant.
Comparison to existing counterexamples to highlight distinct properties between the degree-four member and Alpöge's maps.
Explicit polynomial maps are constructed for each n ≥ 3, confirming the validity of the counterexamples.
The degree-four map is shown to be distinct from previously known degree-three maps under polynomial coordinate transformations.
Demonstrates that constant nonzero Jacobians are not sufficient to guarantee injectivity in dimension three.