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On the Jacobian Conjecture and the Nilpotent Tree Inversion of Keller Mappings

Year: 2026
Type: preprint
Abstract: This preprint provides an exhaustive mathematical monograph on Smale's 16th Problem (Steve Smale, 2000), dedicated to the celebrated Jacobian Conjecture originally posed by Ott-Heinrich Keller in 1939. The conjecture asserts that if a polynomial mapping F = (F1, …, Fn) : &complex;n → &complex;n has a non-zero constant Jacobian determinant det Jac(F)(x) ∈ &complex;* for all x ∈ &complex;n, then F is a global polynomial automorphism of &complex;n (i.e. globally bijective with a polynomial inverse): det Jac(F)(x) = 1 &implies; F ∈ Aut(&complex;[x1, …, xn]) This 11-page treatise presents an algebraic and geometric survey encompassing the Bialynicki-Birula & Rosenlicht injectivity theorem, Pinchuk's counterexample in the real case ℝ2, the Bass-Connell-Wright cubic- nilpotent reduction, Drużkowski matrix Jordan stratifications, the formal tree expansion inverse formula, and the Kontsevich-Dixmier Weyl algebra equivalence. Key Mathematical Results & Contributions Axiomatic Formulation & Dimension 1 Proof: Rigorous proof that in dimension n = 1, Keller mappings reduce to degree-1 affine bijections F(x) = ax + b with polynomial inverse F−1(y) = (y − b)/a. Bass-Connell-Wright (1982) & Yagzhev (1980) Cubic Reduction: Complete derivation showing that the Jacobian Conjecture in all dimensions is equivalent to the invertibility of cubic-homogeneous mappings F(x) = x − H(x) where Jac(H) is strictly nilpotent of index n. Drużkowski Cubic-Linear Factorization: Structural analysis of H(x) = (Ax)*3 and Jordan nilpotent strata for the coefficient matrix A. Wright Nilpotent Tree Inversion Formula: Explicit combinatorial sum of rooted differential operator trees proving that the formal inverse series F−1(y) truncates into a finite polynomial of degree at most 3n−1 under Jacobian nilpotency. Kontsevich & Dixmier Equivalence: Detailed exposition of the Kontsevich–Belov-Kanel–Tsuchimoto theorem establishing that the Jacobian Conjecture in dimension 2n is equivalent to the Dixmier Conjecture on the Weyl algebra An(&complex;). 100% Machine-Checked Verification in Lean 4: Invertibility lemmas, triangular polynomial automorphisms, exact quadratic/cubic tree inversion formulas, and nilpotent operator algebras are formally certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/smale-problemsInteractive Portal: https://maths-proofs.edounze.com/#smale-16-jacobian Primary MSC (2020): 14R15, 13B10, 14E07, 16S32, 68V20.Keywords: Jacobian Conjecture, Smale's 16th Problem, Keller Mappings, Polynomial Automorphisms, Bass-Connell-Wright Reduction, Drużkowski Matrices, Nilpotent Tree Inversion, Dixmier Conjecture, Formal Verification, Lean 4, Mathlib.
DOI: 10.5281/zenodo.22133105
Language: English
Open Access status: green
OpenAlex ID: W7204453912
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