Terence Tao and ChatGPT: Deconstructing the Jacobian Conjecture Counterexample

Expert-Led AI Collaboration in Higher Mathematics

Terence Tao, one of the world's leading mathematicians, recently used ChatGPT to analyze a counterexample to the Jacobian Conjecture. The interaction demonstrates a shift in LLM usage: rather than using the AI as a basic query tool, Tao employs it as a symbolic computation engine and a sounding board to map a complex mathematical construction to a simpler, more intuitive framework.

This collaboration highlights that the utility of frontier models in highly specialized fields depends heavily on the user's ability to provide precise, jargon-heavy steering and structured prompts that guide the model through a logical progression of discovery.

Symbolic Verification of the Jacobian Counterexample

In the shared conversation, Tao uses ChatGPT to verify a specific polynomial map and its properties. The core of the technical exchange focuses on proving that a specific variety $X$ is isomorphic to affine space $\mathbb{A}^{3}$.

Proving the Isomorphism $X \cong \mathbb{A}^{3}$

ChatGPT verifies that two maps, $\Phi$ and $\Psi$, are mutually inverse polynomial isomorphisms.

1. The Forward Map $\Phi$: Maps $\mathbb{A}^{3}$ to a variety $X$ defined by the equations $a^{2}e - abd + cb^{2} = 1$ and $ad + bc = 1$.
2. The Inverse Map $\Psi$: Maps $X$ back to $\mathbb{A}^{3}$ using polynomial formulas for $y$ and $z$.

By substituting the formulas and reducing the results modulo the defining equations of $X$, the AI confirms that $\Psi \circ \Phi = (id){\mathbb{A}^{3}}$ and $\Phi \circ \Psi = (id){X}$, establishing the isomorphism.

Constructing the Keller Map

Tao then directs the AI to compose $\Phi$ with a specific map $F(a,b,c,d,e) = (ac, ad + bc, ae + bd, be)$ and drop the constant term $ad+bc=1$. This process yields a polynomial map $G: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ with a constant nonzero Jacobian determinant of $-1$.

This resulting map $G$ is a "Keller map," which is central to the Jacobian Conjecture. The AI symbolically differentiates the map to confirm the determinant is constant, providing a consistency check at $a=0$ to verify the global calculation.

Mapping to the Original Counterexample

Finally, Tao asks the AI to relate this constructed map $G$ to a known counterexample $F_{orig}$. The AI identifies that the two are identical up to linear changes of coordinates and output rescalings:

$$F_{orig} = B \circ G \circ A$$

Where $A$ and $B$ are linear maps. This verification confirms that the factorization construction recovered the original counterexample exactly.

Analysis of the Human-AI Interaction Pattern

Observers of the transcript noted several key patterns in how an expert interacts with frontier models to achieve high-level technical results.

The "Colleague" Model of Interaction

Unlike typical chatbot usage, the interaction is characterized by a peer-to-peer dialogue. As noted by community members, the AI does not act as a tool but as a "colleague," providing analysis, inference, and symbolic verification that Tao relies on to refine his understanding.

High-Precision Steering

The success of the session was not due to the AI's autonomous "intelligence" but to Tao's specific, high-level prompts. The AI was steered using:

• Domain-Specific Jargon: Using precise mathematical terminology to set the model's operational context.
• Incremental Progression: Moving from basic verification to composition, and finally to the mapping of two different forms of the same result.
• Iterative Refinement: Using the AI to "brute force" the symbolic algebra while the human provides the strategic direction.

The "Cognitohazard" of Known Unsolved Problems

One interesting observation from the discussion is that LLMs may be hindered by their training data regarding "unsolved" problems. Because the model "knows" the Jacobian Conjecture is unsolved, it may be predisposed to doubt a user's proof. However, when prompted to perform specific symbolic calculations (like computing a Jacobian), the model's computational abilities can override its probabilistic knowledge of the problem's status, allowing it to verify a counterexample that its own internal knowledge base would otherwise flag as unlikely.