As millions of people recently celebrated the excitement of the FIFA World Cup Final, a different kind of breakthrough was generating excitement in the mathematical community: an artificial intelligence system had helped uncover a counterexample to the famous Jacobian conjecture.
Levent Alpöge, a mathematician at artificial intelligence (AI) company Anthropic, made the understated announcement on X. He said he had discovered a counterexample to the Jacobian conjecture, a longstanding problem in algebraic geometry, with the help of Anthropic’s large language model Claude Fable 5, which had been released publicly only a few weeks earlier.
The discovery is the latest in a series of notable mathematical advances involving large language models. However, this result stands out because the counterexample is remarkably short and straightforward to verify.
What is the Jacobian conjecture?
A conjecture is a mathematical statement that researchers believe may be true but have not yet been able to prove or disprove.
The Jacobian conjecture is an abstract problem in algebraic geometry, but its basic idea can be explained using functions.
Functions work like mathematical machines: they take one or more numbers as inputs and produce other numbers according to a specific rule or equation. In the Jacobian conjecture, these rules are based on polynomials.
The numbers can represent points in space, such as coordinates on a map. A polynomial function can therefore be viewed as a transformation that moves points from one location to another.
Mathematicians can measure how smoothly a function transforms space by calculating its Jacobian determinant. When the Jacobian determinant is a non-zero constant, the function does not locally fold or compress space around any point.
The Jacobian conjecture proposes that if a polynomial function has a non-zero constant Jacobian determinant, it must have an inverse function that is also made up of polynomials. In other words, the original transformation should be reversible, allowing every point to be returned to its starting position.
Not every function is reversible. If two different starting points are mapped to the same destination, for example, there is no way to determine which original point each one came from. The information has been lost, making the function impossible to reverse.
A century of attempts to solve the problem
The two-dimensional version of the Jacobian conjecture was proposed by Czech mathematician Ludwig Kraus in 1884. German mathematician Ott-Heinrich Keller later generalised the problem to any number of dimensions in 1939.
The conjecture was considered important enough to be included by Fields Medallist Stephen Smale in his 1998 list of Mathematical Problems for the Next Century.
Over the years, mathematicians have announced many claimed proofs, including attempts by renowned 20th-century mathematicians Beniamino Segre and Wolfgang Gröbner. However, each argument was eventually found to contain subtle errors that invalidated the proof.
There have also been valid results proving the conjecture under particular restrictions. Computational studies have shown that it holds in two dimensions for polynomial functions with degrees up to 100, meaning the variables can include powers as high as 100.
Until now, however, nobody had proved the conjecture in its full generality or produced a function demonstrating that it was false.
A surprisingly simple counterexample
One reason the Jacobian conjecture has attracted so much attention is that a counterexample should, in principle, be easy to describe. It is simple to create functions that merge multiple points, as well as polynomial mappings with a constant Jacobian determinant.
The challenge is to construct a polynomial mapping with both properties at the same time. As one Math Stack Exchange user observed in a 2017 post, “for all what we know, some smart undergraduate can simply write a formula […] that will be a counter-example to this conjecture”.
That is essentially what happened with Alpöge’s discovery. He found a three-dimensional polynomial function with a constant Jacobian determinant of -2 that maps multiple input points to the same output point. Because the transformation is not reversible, it provides a counterexample to the Jacobian conjecture.
The result shows that the conjecture is false in every dimension greater than two. The original two-dimensional version, however, remains unsolved. Because Alpöge’s counterexample is so concise, other mathematicians were able to verify it relatively quickly.
AI and the search for new mathematical ideas
Alpöge’s discovery is the latest in a growing list of high-profile mathematical breakthroughs involving large language models. Recent examples include OpenAI’s disproof of the unit distance conjecture and the proof of Erdős’ problem 1196 by Liam Price, a 23-year-old amateur mathematician.
These examples highlight one of the most powerful capabilities of AI systems: they can combine concepts from different areas of mathematics and use those connections to produce unexpected arguments and solutions.
At the time of writing, the precise prompts Alpöge used and the AI model’s original output have not been made public. Even so, the Jacobian conjecture result appears to differ from many other recent AI-assisted mathematical achievements.
In this case, the counterexample is not based on a complicated construction or an especially long proof. Instead, the main challenge appears to have been searching through an enormous number of possible polynomial mappings to find one with the required properties.
This suggests that AI could become valuable not only for constructing mathematical proofs but also for discovering unexpected mathematical objects and counterexamples. What that will mean for the future of mathematics and the role of human mathematicians remains an open question.
Source: www.sciencedaily.com