A Valuation–Theoretic Approach of the Jacobian Conjecture in Dimension Two

I present a valuation–theoretic approach to the Jacobian Conjecture in dimension two over algebraically closed fields of characteristic zero. Starting from a polynomial endomorphism of the affine plane with constant nonzero Jacobian determinant, we pass to the normalization of the projective graph and reduce the problem via Stein factorization to a finite morphism. A fiber–length argument eliminates horizontal boundary divisors, showing that any potential ramification must occur over the affine locus. Assuming such affine ramification, we analyze the corresponding dicritical divisor using discrete valuation theory. A Rees degeneration to the associated graded ring transports transcendence information to the special fiber and forces a rank–one structure in the valuation lattice. This yields a multiplicative character relation among the leading terms, which in turn implies strict positivity of the valuation of the Jacobian two–form. This contradicts the Keller identity and eliminates affine ramification. Combined with the triviality of finite étale covers of the affine plane in characteristic zero and Zariski’s Main Theorem, this establishes that the original polynomial morphism is an automorphism.