The Jacobian Conjecture via Flag Reduction and 'Etale Fundamental Groups

摘要: We prove the Jacobian Conjecture: every polynomial map F:\CCn→\CCnwith constant nonzero Jacobian determinant is a polynomial automorphism.
By the Drużkowski reduction (2001), it suffices to consider maps F(x)=x+(Ax)∗3with A2=0and det(JF)=1. Under this reduction, we establish the Flag Reduction Theorem: the condition A2=0forces im(A)⊆ker(A), yielding a flag basis in which Fdecomposes as F(xk​,xw​)=(xk​+p(xw​),h(xw​))with det(JF)=det(Jh)=1.
The reduced map h:\CCr→\CCr(where r=rank(A)≤n/2) satisfies det(Jh)=1as a constant. We prove his an isomorphism through the following chain: constant Jacobian ⇒quasi-finite (positive-dimensional fibers force det=0at smooth points) ⇒flat (Miracle Flatness: \CCrregular, Cohen-Macaulay, zero-dimensional fibers) ⇒étale (flat + unramified) ⇒finite étale cover of \CCr(Zariski’s Main Theorem) ⇒degree 1 (π1eˊt​(\CCr)=0by SGA1 comparison) ⇒isomorphism (normality + unit group argument). Reconstruction of F−1from h−1then yields a polynomial inverse.
We provide a detailed comparison with Oda’s related approach (arXiv:0706.1138, 99 versions), identifying three fatal defects: circular definition of simple connectivity, conflation of unramified with étale (missing flatness), and assumption of finiteness. Our proof avoids all three. We also clarify how our result completes the Bass–Connell–Wright (1982) covering space route by establishing finiteness from the Keller condition under the A2=0structure.
Computational verification (exact arithmetic, n≤8) confirms all structural claims with zero counterexamples across more than 17,000 Keller matrices.