# An Analysis of the Jacobian Conjecture Counterexample

The refutation of a mathematical conjecture and the role of AI.

- **Source:** [Hacker News](https://news.ycombinator.com)
- **Published:** 2026-07-21
- **Topic:** [Data & AI](https://brifing.si/c/data-ai)
- **Original article:** <https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/>
- **On Brifing:** https://brifing.si/s/d9cdff17-302b-4ef4-968c-540f9ad95577

## What is the Jacobian Conjecture?

The Jacobian conjecture is a mathematical claim that local invertibility of polynomial maps in complex variables implies global invertibility. This conjecture has long been a topic of debate among mathematicians.

## Basic Formulation of the Conjecture

The conjecture states that if the Jacobian determinant of a polynomial map is a non-zero constant, then the map has a polynomial inverse. This implies that local invertibility guarantees global invertibility.

## Discovery of the Counterexample

Recently, using Fable AI, it was shown that a three-dimensional polynomial map with a constant Jacobian determinant is not invertible. This proves the conjecture invalid in three and higher dimensions.

## Details of the Counterexample

The example involves a seventh-degree polynomial. Calculation of the Jacobian determinant reveals that all non-constant coefficients vanish, resulting in an unexpected massive cancellation of terms.

## Geometric Explanation

The counterexample was later explained in geometric terms. This explanation relies on algebraic geometry to understand the behavior of polynomial maps and shows why the conjecture fails in higher dimensions.

## Local Injectivity Formulation

The theorem was reformulated using local injectivity. It shows the existence of an affine variety, isomorphic via polynomial changes, and a polynomial map that is locally injective but not globally injective.

## Role of the Multiplication Map

The solution was derived using the multiplication of low-degree polynomials. The product of linear and quadratic polynomials forms a map producing cubic polynomials, which interacts with a large symmetry group.

## Symmetries and Normalization

The multiplication map has a five-dimensional symmetry group. These symmetries affect the injectivity of the map. Scaling symmetry is used to normalize the map over a four-dimensional variety.

## Concept of Resultant

The resultant measures whether linear and quadratic polynomials share a common root. This concept behaves compatibly with symmetries and is used in normalizing the scaling symmetry.

## Injectivity of the Map

The normalized multiplication map goes from a four-dimensional variety to another. However, this map is generally three-to-one non-injective, a critical step in refuting the Jacobian conjecture.

## Mathematical Implications

This counterexample definitively proves the Jacobian conjecture invalid in higher dimensions. It also highlights the potential of AI-assisted mathematical discoveries.

## Broader Significance

This discovery underscores the role of AI in validating mathematical conjectures. It also paves the way for new geometric methods to understand polynomial map behavior.

## Sources covering this story

- https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/ — 

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_Summary written by Brifing from the original reporting. Copyright in the original article belongs to its publisher; read the full piece at the link above._