KellerMap

A Python library for polynomial Keller maps — polynomial endomorphisms with a nonzero constant Jacobian determinant — and for certified transformations of them.

Whether every Keller map is an automorphism was the Jacobian conjecture, posed by Ott-Heinrich Keller in 1939 — or, on a recent reading of the sources, by Ludwig Kraus in 1884; see docs/references.md. It stood until July 2026 and is now known to be false in every dimension n >= 3. The maps this library is built to handle are therefore precisely those that need not be invertible, which is why it is named after them rather than after automorphisms.

There is a local coincidence in that name. Keller held the second chair of mathematics at the Martin-Luther-Universität Halle-Wittenberg from 1951 until his emeritation in 1971, succeeding Heinrich W. E. Jung, and he lectured there into the 1980s — the present-day academic home of this project. Sources differ on the year; docs/references.md says which and why.

The first goal is the degree reduction introduced by

H. Bass, E. H. Connell, D. Wright, The Jacobian Conjecture: Reduction of Degree and Formal Expansion of the Inverse, Bull. Amer. Math. Soc., 1982.

The library follows the reduction past that paper. Prellberg's collision-hull compression and the symmetric lift of de Bondt and van den Essen take the cubic homogeneous map to the gradient of a quartic form. By Prellberg, arXiv:2608.12543v3, that form is a counterexample to Zhao's Vanishing Conjecture, and VanishingWitness holds it together with the collision that makes it one. docs/references.md gives the sources.

The implementation is intended as both

• a research tool for experiments related to the Jacobian Conjecture,
• and a faithful software implementation of the original mathematical proof.

Unlike a simple symbolic manipulation package, every reduction step is represented explicitly as a mathematical object and carries its own machine-checkable certificate.

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Project Status

Current version: 0.8.0

DOI: 10.5281/zenodo.23295965. That is the DOI of the newest release, reserved on Zenodo before that release. A release candidate is not deposited and has none of its own, so while the candidates for a release run, the DOI names the release before them: 0.6.0 through the candidates for 0.7.0, and 0.7.0 through those for 0.8.0. CITATION.cff carries the same number and a test holds the two against each other. The sentence used to name the candidate it was written for and went stale at the next one, which is why it names none.

Concept DOI: 10.5281/zenodo.22299351. That one resolves to the newest version. Cite it for the software, and the version DOI above for a particular state of it. docs/deposit.md says what the record holds.

What the library does

• Polynomial maps over a sparse PolyRing, with value semantics: composition, extension, reordering, Jacobian matrices and determinants.
• Elementary and linear automorphisms, EA_n(k) and GL_n(k), the latter as an ordered product of Gauss operations.
• Certified steps. BCWStep is one application of Bass–Connell–Wright, Proposition (3.1), that verifies rather than asserts: it keeps the factorization it was given and checks it, and a failure names the obligation it broke. LinearStep and TranslationStep are the two factors of the linear normalization.
• The second and third stages of the Reduction Theorem. UnipotentStep is Section 4's second step, which doubles the dimension and makes the Jacobian of the displacement nilpotent; HomogenizationStep is the third, which adds one variable and makes the displacement cubic homogeneous. Theorem 2.1(b) asks in addition for a form linear in every variable except the homogenizing parameter; reduce_to_multi_affine is what reaches it, and HOM-11 and HOM-12 are what say so at the end of the chain.
• Compression and the gradient form. CompressionStep restricts a homogeneous map to the subspace its collision generates; SymmetricLiftStep turns the result into the gradient of a quartic over k(i), which is the object Zhao's Vanishing Conjecture is about. VanishingWitness holds that quartic with a collision of its gradient map and checks what a finite computation can check about the conjecture.
• The descent. DescentStep deletes a coordinate that two elementary changes of determinant one have made triangular. It is the second of the two step types that lower a dimension, and the only one whose target may have a higher degree than its source. It verifies a claim a caller supplies rather than looking for the two changes itself.
• Chains. Reduction joins steps and checks the adjacency; ReductionContext checks that a naming policy stays consistent along one.
• Collisions. Collision is the evidence that a map is not injective, and it is transported across every step, so a reduction of a counterexample is still a counterexample. Two of the eight step types may refuse a collision rather than carry it, and say why.
• Four searches. search walks from a source towards a target and is told what a fresh coordinate may carry; peel walks back from a target and is told nothing else; reduce_to_degree3 is given a source alone and reduces it to degree three; reduce_to_multi_affine starts there and removes the squares, which is the half of Theorem 2.1(b) the homogenization cannot supply.
• Example maps that recur, including four maps this project did not write and the reductions it derived from them.
• Obligations, not conventions. Every promise the verification surface makes is written in docs/contracts.md under a stable identifier, and the exception that fails cites it.

This milestone, 0.8

The last link of the chain this project follows: Jacobian Conjecture, BCW reduction, gradient form, Zhao's Vanishing Conjecture.

The witness. VanishingWitness holds the gradient form of a symmetric lift together with a collision of its gradient map. By Theorem 3, part 4, of arXiv:2608.12543v3 that form is a counterexample to Zhao's Vanishing Conjecture. verify() checks the lift, the collision, the hypothesis Delta^m(P^m) = 0 up to a depth, and Delta(P^2) != 0. On the forty-variable form of the paper, the library and a reconstruction script that does not use it reach the count of Delta(P^2) that the paper states, by two different methods. docs/milestones.md under "Version 0.8" has the figures.

The conclusion is derived and not computed. A non-zero value of Delta^m(P^(m+1)) at any index above a known bound would decide it, and for the forms here that bound is far beyond any computation. VAN-5 says so.

The scripts. A script whose data the source archive does not carry now says that it is not checked, instead of failing. Every measurement prints the machine, the date and the state of the tree. make benchmark records counts that do not depend on the machine, so that the cost of the pipeline can be compared between releases.

Next, 0.9

Certificates outside the running process. Today a certificate exists only as an object in memory, and it is checked independently only by eight reconstruction scripts written by hand, one per chain. Milestone 0.9 gives a certified chain and its witness a file format, stated as obligations first, and an exporter. An independent checker that does not import the library checks such a file. The eight reconstructions stay as its controls. A LaTeX export is the first view of a certificate, and a small command line runs both.

The research questions milestone 0.7 left open go to a milestone of their own. The plan is in docs/roadmap.md. CHANGELOG.md lists what each release changed, and the milestones before this one are there rather than here.

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Installation

pip install kellermap

Requires Python 3.10 or newer and SymPy 1.14 or newer.

Quick start

import sympy as sp
from kellermap import PolynomialMap

x, y = sp.symbols("x y")
F = PolynomialMap((x, y), (x + y**3, y))

F.determinant()  # 1 — a Keller map
F.degree()  # 3
F.filtration_degree()  # 2, from ord(F - X) = 3
F.extend(2).variables  # (x, y, X3, X4)

The first two steps of the reduction that this library exists for — Alpöge's counterexample to the Jacobian conjecture, on its way down to degree three:

import sympy as sp
from kellermap import Collision, PolynomialMap, Reduction, ReductionContext
from kellermap import over_field
from kellermap.bcw import BCWStep, Fresh
from kellermap.reduction import LinearStep

x1, x2, x3 = sp.symbols("x1 x2 x3")
R = sp.Rational

alpoege = over_field(
    PolynomialMap(
        (x1, x2, x3),
        (
            (1 + x1 * x2) ** 3 * x3 + x2**2 * (1 + x1 * x2) * (4 + 3 * x1 * x2),
            x2 + 3 * x1 * (1 + x1 * x2) ** 2 * x3 + 3 * x1 * x2**2 * (4 + 3 * x1 * x2),
            2 * x1 - 3 * x1**2 * x2 - x1**3 * x3,
        ),
    )
)
collision = Collision.at(
    alpoege,
    (
        (0, 0, R(-1, 4)),
        (1, R(-3, 2), R(13, 2)),
        (-1, R(3, 2), R(13, 2)),
    ),
)

normalization = LinearStep.normalize(alpoege)  # F_(1)^-1 o F, BCW II (1.1)
u, v = ReductionContext().variables(normalization.target.ring, 2)
first = BCWStep.build(  # Proposition (3.1): two new dimensions
    normalization.target,
    0,
    Fresh(-x1 * x3 / 2, u),
    Fresh(x1**2, v),
)
reduction = Reduction([normalization, first])

reduction.verify()  # None, or VerificationError naming the obligation that failed
reduction.dimensions()  # (3, 3, 5)
reduction.transport(collision).points[1]  # (1, -3/2, 13/2, 13/4, -1)

The counterexample is still a counterexample at the other end, and the chain says so by carrying it rather than by asserting it.

docs/api.md covers the rest; every example in it is executed by the test suite, as are both blocks above.

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How this project is built

The code in this repository is written in collaboration with a large language model, Anthropic's Claude. AGENTS.md at the repository root holds the working agreements: how a change is delivered, what a certificate is for, how claims and sources are handled, and where the assistant is expected to push back rather than comply. It is the same file the assistant works from.

The maintainer takes responsibility for everything here, whoever or whatever produced a first draft of it. Each milestone since 0.2 has gone through external audits before release, and every finding is recorded.

This is worth one distinction, because the two halves of the repository carry different kinds of assurance.

The mathematics is machine-checkable and is checked. A reduction is a chain of certified steps, each obligation has a number in docs/contracts.md, and the exception that fails cites it. The reduction of the published nineteen-dimensional map exists three times over by routes that share no code: as a chain of verified steps, as an independent computation in plain SymPy, and as a search result. scripts/mutation_probe.py breaks one promise at a time to ask whether the suite would notice. None of that asks anyone to trust the producer, which is the point of building it that way.

The prose is not machine-checkable in the same sense. Page numbers, licences and statements about the literature rest on somebody having opened the source. Two errors of that kind were made during 0.4 and both were found by reading a scan page by page: a citation of the filtration MA_n^d(k) gave p. 304 where it is p. 303, and the licence of arXiv:2608.00222 was recorded as undeterminable when the listing states it. docs/references.md records what each claim rests on, and tests/test_documentation.py checks what the prose says about the code, which is the part of it that can be checked.

If you want to contribute, CONTRIBUTING.md says how, including what is expected of AI-assisted contributions.

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Documentation

The documentation is located in

docs/

Most important documents:

api.md            public API, with examples the test suite executes
architecture.md   design decisions and the reasons for them
contracts.md      binding obligations of the verification surface
references.md     sources, and the comparisons with published figures
provenance.md     where the fixed data came from, and how this repository
                  was written
errata.md         what this project reported wrongly, and the correction
deposit.md        what goes into the Zenodo record, and why not the
                  automatic route
roadmap.md        the open milestone and the long-term goals
milestones.md     the plans of the released milestones

CHANGELOG.md and CITATION.cff sit at the repository root.

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License

MIT License