You may have heard about a new mathematical result announced by Levent Alpöge, developed using Fable a few days ago, concerning the Jacobian conjecture. Unfortunately, many discussions of the result merely restate the conjecture's formal definition, offering little in the way of an accessible explanation.
This is understandable: mathematicians use precise, formal language that can be impenetrable to a general audience. Explaining every technical term from scratch would take as long as earning a math PhD. However, this particular result can be understood without that level of training. While some precision is lost in the simplification, the core idea becomes much clearer. The explanation still requires some math, but at the level of a typical STEM undergraduate rather than a specialized mathematician.
The Jacobian conjecture asks whether a certain type of function must always be invertible. The new disproof provides a counterexample: a function that meets all the conjecture's requirements but provably cannot be inverted.
As a reminder, a function is invertible only if its output uniquely determines its input. If a function produces the same output for two different inputs, it cannot be reversed. While other conditions are also required for invertibility, this necessary one-to-one property is fundamental. A function lacking this property cannot be inverted.
Therefore, if we can find a function that satisfies the conjecture's criteria but is not one-to-one, we have a contradiction: the function matches the conjecture's description yet cannot be invertible, so the conjecture is false.
The formal conjecture applies to polynomial functions over any "field of characteristic zero." However, the counterexample involves ordinary real numbers in three-dimensional space. This allows us to explain the concept using everyday three-dimensional intuition.
Imagine an infinitely large, infinitely stretchy block of rubber filling the entire universe. While physically impossible, we can conceptualize it. Now, imagine deforming this block by stretching, bending, twisting, or otherwise reshaping it. Each point in the rubber begins at some (x, y, z) location, and we move it to a new (x, y, z) location defined by a function. This function, often called a deformer or deformation function, tells us where each point goes—a common concept in physics and computer graphics.
If we allowed any arbitrary function, it could do harmful things to the rubber: tear apart neighboring points, create sharp creases, or collapse an entire volume into a degenerate surface, line, or single point. The Jacobian conjecture prohibits such behaviors. It requires that the function's "Jacobian determinant" be constant and nonzero—a term that essentially means the deformation preserves local volume everywhere, without tearing or collapsing.
In simpler terms, the conjecture suggests that if a polynomial function deforms space smoothly and without compressing any region to zero volume, then it must be reversible: you can always compute the original position of each point. The counterexample shows that this is not always true, providing a function that meets the smoothness and volume-preserving conditions yet is not one-to-one, thus disproving the conjecture.
As of 2026, this result remains unverified by the broader mathematical community, but it has sparked significant interest and debate. If confirmed, it would resolve a longstanding open problem in algebraic geometry and polynomial mapping theory.