A quartic-resultant counterexample to the Jacobian Conjecture

Paper (PDF) · LaTeX source · Claim and evidence ledger · Resumen en español

This repository accompanies a preliminary research note on an explicit noninjective Keller map

[ H:\mathbb C^4\longrightarrow\mathbb C^4 ]

with constant Jacobian determinant -2. The construction is organized by a normalized factorization of a quartic into a linear factor and a cubic factor. Its generic fiber has eight points, grouped into four natural pairs.

Main result

The two finite identities needed for the counterexample are

[ \det J_H=-2 ]

and

[ H(1,-1,13/2,0)=H(-1,1,-13/2,0)=(0,0,1,0). ]

The two source points are distinct; therefore, H is a counterexample to the Jacobian Conjecture in dimension four. Both identities are checked exactly, without floating-point arithmetic, by two independent implementations.

Scope and status

This is a preliminary research release dated 21 July 2026.

• The constant-Jacobian and collision certificates are exact and machine checked.
• The factorization identities and generic degree eight are proved in the note.
• The order-192 imprimitive monodromy is accompanied by a square-class proof.
• No claim of historical novelty or priority is made pending external review.
• No claim is made that coupling a fourth variable, by itself, rules out equivalence to a suspension under polynomial coordinate changes.

The claim and evidence ledger records the precise evidence level of each major statement and lists the claims that are deliberately not being made.

The map

In coordinates (x,y,z,r), set

[ u=1+xy, ]

[ a=-1+2xy-\frac72x^2y^2+\frac12x^2r+x^3z, ]

[ b=y+\frac12xr-\frac32xy^2+x^2z+\frac12x^2yr -\frac72x^2y^3+x^3yz, ]

[ \begin{aligned} c={}&2z+2yr-11y^3+6xyz+3xy^2r-17xy^4\ &+6x^2y^2z+x^2y^3r-7x^2y^5+2x^3y^3z. \end{aligned} ]

Then

[ H(x,y,z,r)=(xa,;xr+ub,;xc-ur,;uc). ]

Reproducibility

The dependency-free checker uses only the Python standard library:

python verification/exact_sparse.py

The independent SymPy checker redefines the map from scratch and also verifies the factorization, the root-evaluation and SymPy resultant values, the universal six-variable Jacobian, and an explicit eight-point fiber:

python -m pip install -r requirements-dev.txt
python verification/sympy_verify.py

GitHub Actions verifies the artifact manifest and runs both mathematical checks on every push and pull request. The committed checksums in CHECKSUMS.sha256 identify the exact research artifacts in this version; they can be checked locally with python verification/check_manifest.py.

To regenerate the typeset PDF:

python tools/build_pdf.py
python tools/build_pdf.py --check

This requires Tectonic 0.16.9 on PATH (or in the TECTONIC environment variable). The builder invokes only paper/manuscript.tex, fixes the TeX support bundle and source date, and can compare the result byte for byte with the committed PDF.

Repository layout

• paper/manuscript.tex - source of the preliminary note.
• paper/manuscript.pdf - typeset paper for direct reading and download.
• tools/build_pdf.py - reproducible TeX-to-PDF build and consistency check.
• verification/exact_sparse.py - dependency-free exact certificate.
• verification/sympy_verify.py - independent computer-algebra audit.
• verification/check_manifest.py - dependency-free artifact-integrity check.
• notes/construction.md - derivation and possible generalization.
• CLAIMS.md - claim/evidence ledger and explicit non-claims.

Resultant normalization

For a degree-one polynomial f and a degree-three polynomial g, the note uses

[ \operatorname{Res}*(f,g)=a_f^3\prod{f(\alpha)=0}g(\alpha). ]

With this normalization, Res_*(L,Q)=1. SymPy 1.14 reports resultant(L,Q)=-1 for the same displayed factors. The checker records both values explicitly; only the fact that the resultant is a nonzero constant is needed for the construction.

Citation and attribution

The preliminary release is attributed to the stable public pseudonym juanmgh3. Authorship and computational assistance are documented in AUTHORS.md.

License

Code is released under the MIT License. The manuscript and explanatory text are released under Creative Commons Attribution 4.0. See LICENSE and LICENSES/CC-BY-4.0.txt, respectively. Copyright 2026 juanmgh3.