Algebroids and Jacobian conjecture

Using the Galois theory over function field, and the holomorphy of algebroids defined via irreducible polynomial at singular points, we prove the injectivity of any kellerian mapping. The famous Jacobian conjecture is true.

💡 Research Summary

The paper titled “Algebroids and Jacobian conjecture” claims to settle the long‑standing Jacobian conjecture by proving that every Kellerian (i.e., polynomial) map with constant Jacobian determinant equal to one is injective, and therefore bijective, on (\mathbb{C}^n). The author’s strategy is to combine two seemingly unrelated tools: Galois theory over function fields and a newly introduced notion of “algebroid” – a multi‑valued analytic function defined locally by an irreducible polynomial equation at singular points.

The first part of the argument establishes a Galois‑theoretic bridge between the rational function field (\mathbb{C}(x_1,\dots,x_n)) and the subfield generated by the components of a Keller map (F=(F_1,\dots,F_n)). By invoking the classical result that the degree of the field extension (