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Collision-Hull Compression for Homogeneous Keller Maps and a Forty-Variable Counterexample to Zhao's Vanishing Conjecture

This paper introduces a collision-generated compression principle for homogeneous Keller maps that canonically recovers known dimension reductions and provides an explicit 40-variable counterexample to Zhao's Vanishing Conjecture over $\mathbb{Q}(i)$ .

Original authors: Thomas Prellberg

Published 2026-08-14

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CC BY 4.0

Original authors: Thomas Prellberg

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a massive, invisible puzzle that has stumped the world's best math detectives for decades. This puzzle lives in the strange, twisting world of algebraic geometry, a branch of mathematics that studies shapes defined by equations. The specific mystery at hand is called the Jacobian Conjecture. Think of it as a rule about how shapes can be squished, stretched, or twisted without ever tearing or folding over themselves. The rule says: if you have a specific kind of mathematical machine (a polynomial map) that preserves a certain "volume" of space, then that machine must be reversible—you should always be able to run the machine backward to get exactly where you started.

For a long time, no one could prove this rule was true for every possible size of machine. However, a mathematician named Wenhua Zhao proposed a clever shortcut. He suggested that if you can find a specific type of "broken" machine—a machine made of a four-part (quartic) equation that behaves like a nilpotent (a shape that eventually flattens to nothing when you keep squishing it) and cannot be reversed—then the whole big rule would be proven false. This is the Vanishing Conjecture: a challenge to find a specific, broken machine that looks perfect on the surface but secretly fails the test. If such a machine exists, the big rule is wrong. If it doesn't exist, the rule might be true. The stakes are high because this puzzle connects to how we understand the fundamental structure of space and equations.

Now, enter Thomas Prellberg, who has built a very specific, very large "broken machine" to test this idea. The paper doesn't claim to have solved the entire puzzle for everyone, but it has constructed a massive, 40-dimensional example that breaks Zhao's specific test.

Here is how the story unfolds. Prellberg started with a known, smaller machine created by a mathematician named Thompson. Thompson's machine had 24 moving parts (variables) and was a "cubic" machine (built from three-part equations). It was already known to have a "collision": two different starting points that ended up in the exact same spot, proving the machine couldn't be reversed. However, Thompson's machine was too messy to use directly for Zhao's test.

Prellberg used a technique called Collision-Hull Compression. Imagine you have a tangled ball of yarn (the 24 variables) and you want to find the smallest, tightest knot that still holds the two tangled ends together. Prellberg showed that if you take the two points that collide and keep mixing them together using the machine's rules, you eventually generate a smaller, tighter space. In Thompson's case, this process naturally shrank the 24 variables down to exactly 20. This wasn't a guess; it was a mathematical inevitability. The paper proves that this 20-variable space is the smallest possible container that can hold the collision. Any attempt to squeeze it into 19 or fewer variables would break the collision, meaning the machine would stop working as a counterexample.

Once he had this perfect 20-variable machine, Prellberg applied a "symmetric lift." Think of this as taking a 2D drawing of a cube and folding it into a 3D object, but in math, it doubles the dimensions. He turned the 20-variable cubic machine into a 40-variable quartic (four-part) machine. This new machine, which has exactly 350 monomials (the individual building blocks of the equation), is the star of the show.

The paper proves three critical things about this 40-variable machine:

It is a Hessian-nilpotent polynomial, meaning it has the specific "flattening" property Zhao's test requires.
It satisfies the "vanishing" condition for many steps (mathematically, $\Delta^m P^m = 0$ for all $m \ge 1$ ), making it look like it should work.
Crucially, it fails the final test: the sequence does not stay zero forever. The paper proves that for infinitely many steps, the result is not zero ( $\Delta^m P^{m+1} \neq 0$ ).

Because it fails this final test, the machine is a valid counterexample to Zhao's Vanishing Conjecture. It proves that the specific shortcut Zhao proposed does not work; you cannot simply assume that if the early steps vanish, the machine is safe. The machine is "broken" in the exact way Zhao's hypothesis tried to rule out.

The paper is very careful about what it claims. It does not say it has found the smallest possible counterexample in all of mathematics. In fact, it acknowledges that other researchers have found 38-variable examples using different methods. Instead, the paper's main victory is route-specific minimality. It proves that if you start with Thompson's specific 24-variable machine and try to shrink it down to make a counterexample, you cannot go below 20 variables before applying the lift. The 40-variable result is the smallest you can get via this specific path.

The author, Thomas Prellberg, has been extremely rigorous. The entire calculation, involving massive matrices and complex fractions, was checked by a computer program using exact arithmetic (no rounding errors). The code is even published alongside the paper so anyone can run it and see the math for themselves. The paper concludes that while this 40-variable machine doesn't solve the entire Jacobian Conjecture, it definitively breaks the specific "Vanishing" rule Zhao proposed, showing that the path to solving the big puzzle is more winding and tricky than previously hoped.

Technical Summary: Collision-Hull Compression for Homogeneous Keller Maps

Problem Statement
The paper addresses the Jacobian Conjecture and, more specifically, Zhao's Vanishing Conjecture. Zhao proved that the Jacobian Conjecture is equivalent to the statement that for a homogeneous quartic polynomial $P \in \mathbb{C}[z_1, \dots, z_n]$ , if $\Delta^m P^m = 0$ for all $m \geq 1$ (where $\Delta$ is the Laplacian), then $\Delta^m P^{m+1} = 0$ for all sufficiently large $m$ . A counterexample to this conjecture requires a homogeneous Hessian-nilpotent quartic $P$ such that the map $id - \nabla P$ is non-injective. While previous work (Alpöge, Speyer, Gao) established the existence of underlying three-variable counterexamples, and Thompson provided an explicit 24-variable cubic-homogeneous Keller map with a rational collision, the construction of a corresponding explicit quartic counterexample with minimal dimension via standard symmetric lifts remained a focus of inquiry.

Methodology
The author introduces a "collision-generated compression principle" within the framework of polarization algebras. The core methodology involves:

Collision Hull Construction: Given a homogeneous Keller map $F = id + h$ with a collision (distinct points $p, q$ such that $F(p)=F(q)$ ), the author constructs the smallest linear subspace $W$ containing $p$ and $q$ that is invariant under the polarization algebra of $h$ . This subspace is generated iteratively by taking the span of the collision points and their images under the symmetric multilinear polarization $T$ of $h$ .
Invariant Restriction: It is proven that this generated subspace $W$ supports a restricted Keller map $g = id + \bar{h}$ which retains the collision and maintains the Keller property (determinant of the Jacobian is a nonzero constant).
Symmetric Lift: The restricted map on $W$ is then subjected to the standard symmetric lift (de Bondt and van den Essen construction), which associates a homogeneous quartic $P_W$ in $2 \dim(W)$ variables to the cubic map on $W$ .
Verification: The paper applies this mechanism to Thompson's 24-variable cubic map, explicitly calculating the collision hull dimensions and verifying the properties of the resulting 40-variable quartic.

Key Contributions and Results

Canonical Recovery of the 20-Dimensional Subspace: For Thompson's original 24-variable cubic map, the collision-generated invariant hull is shown to be exactly MacFarlane's previously identified 20-dimensional invariant subspace. The growth profile of the collision hull dimensions is calculated exactly as $2, 4, 11, 20, 20$. This demonstrates that the known 24-to-20 compression is canonically recovered from the collision itself, without ad hoc selection of invariants.
Explicit 40-Variable Counterexample: The paper constructs an explicit homogeneous quartic $P \in \mathbb{Q}(i)[x_1, \dots, x_{20}, y_1, \dots, y_{20}]$ $P \in Q (i) [x_{1}, \dots, x_{20}, y_{1}, \dots, y_{20}]$ with exactly 350 monomials. This polynomial satisfies:
- The Hessian matrix $Hess(P)$ is nilpotent.
- $\Delta^m P^m = 0$ for all $m \geq 1$ .
- $\Delta^m P^{m+1} \neq 0$ for infinitely many $m$ , thereby providing a counterexample to Zhao's Vanishing Conjecture.
- The map $id - \nabla P$ is non-injective, with two explicit distinct points in $\mathbb{C}^{40}$ mapping to the same image.
Minimality of the Construction Route: The paper establishes that among constructions obtained by restricting Thompson's map to an invariant linear subspace containing the collision and applying the symmetric lift without further alteration, 40 variables is the minimal dimension. This holds over $\mathbb{Q}$ , $\mathbb{Q}(i)$ , and $\mathbb{C}$ .

Significance and Claims
The paper explicitly claims modesty regarding global minimality records. It acknowledges that a 38-variable counterexample exists in other literature (based on a different coordinate reduction) and that the 40-variable lift itself was previously noted by Van Rijn.

The primary significance claimed is structural and canonical:

The work demonstrates that the specific 20-dimensional invariant subspace found by MacFarlane is not an arbitrary choice but is the unique "collision hull" generated by the collision points in Thompson's map.
It provides the first explicit, verified 350-monomial quartic counterexample derived directly from this canonical compression route.
It confirms that no proper invariant linear subspace containing the collision exists prior to the full 20-dimensional space, justifying the dimension of the resulting 40-variable lift within this specific methodological framework.

All principal calculations, including the 350-monomial count and the collision verification, are checked by an accompanying exact arithmetic script using SymPy over $\mathbb{Q}$ and $\mathbb{Q}(i)$ .

Technical Summary: Collision-Hull Compression for Homogeneous Keller Maps

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