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Polynomial contractions of $\mathbb C^d$ and degree growth

This paper constructs a polynomial contraction automorphism of $\mathbb{C}^d$ (for $d \ge 3$ ) with unbounded degree growth, thereby providing an example of an algebraic automorphism that is holomorphically linearizable but not algebraically linearizable.

Original authors: Dmitrii Korshunov

Published 2026-05-29

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Dmitrii Korshunov

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 have a magical machine that takes a point in space and moves it to a new location. If you keep pressing the "run" button on this machine over and over again, the point eventually gets sucked into a single, tiny black hole at the center (the origin). In math terms, this is called a contraction.

Now, imagine this machine is built out of simple rules (polynomials). Usually, when you run these simple rules repeatedly, the complexity of the machine's instructions stays manageable. It's like a recipe that, no matter how many times you follow it, never gets more than a few pages long.

The Big Question
The author of this paper, Dmitrii Korshunov, asked a specific question: Is it possible to build a contraction machine in 3D (or higher) space where the instructions get infinitely more complicated every time you run it?

Think of it like a story.

In 2D (Flatland): If you have a story that always ends with the hero returning home, the plot can only get so wild before it loops back. The paper proves that in 2D, the "story length" (mathematical degree) is always limited. You can't make the instructions infinitely complex.
In 3D (and beyond): The author says, "Yes, we can!" He built a specific machine where, every time you run it, the instructions get slightly longer and more twisted. After a million runs, the instructions are a million pages long. The complexity grows without bound.

How the Machine Works
The author created a specific 3D machine with three dials: $x$ , $y$ , and $z$ .

It shrinks everything slightly (using numbers like 0.1 or 0.2).
But here's the trick: It mixes the dials in a sneaky way. The new position of $x$ depends on $y$ plus the product of $x$ and $z$ .
Because $x$ and $z$ are multiplied together, every time the machine runs, it creates a new "layer" of complexity. It's like a snowball rolling down a hill that keeps picking up more snow, but in a way that the snowball never stops growing in size, even though it's rolling toward a stop.

The "Double Life" of the Machine
This is where the paper gets really interesting. The author shows that this machine has two different "personalities":

The Smooth Personality (Holomorphic): If you look at the machine through a "smooth" mathematical lens (ignoring the fact that it's made of simple algebraic rules), it behaves exactly like a simple, straight-line machine. You can stretch and twist space smoothly to make it look like a simple linear map. This is guaranteed by a famous rule called the Poincaré–Dulac theorem.
The Rigid Personality (Algebraic): However, if you try to fix the machine using only "rigid" algebraic rules (polynomials), you cannot make it look like a simple straight-line machine. The complexity keeps growing, so it can never be simplified into a basic form using these rules.

The Takeaway
The paper proves a surprising fact about the geometry of 3D space (and higher dimensions):

You can have a system that looks perfectly simple and linear if you are allowed to bend space smoothly.
But if you are forced to use only rigid, polynomial rules, that same system is actually incredibly complex and can never be simplified.

In short: In 3D, a machine can be "smoothly simple" but "algebraically messy." This answers a question posed by other mathematicians and shows that the rules of 2D space don't always apply when you add a third dimension.

Technical Summary: Polynomial Contractions of $\mathbb{C}^d$ and Degree Growth

Problem Statement
This paper addresses a question posed by Verbitsky and Ornea regarding the behavior of polynomial contraction automorphisms on complex spaces $\mathbb{C}^d$ . Specifically, it investigates whether there exists a polynomial contraction $\gamma$ (an automorphism with a unique fixed point at the origin such that $\lim_{n\to\infty} \gamma^n(x) = 0$ for all $x$ ) where the degree of the iterates $\deg(\gamma^n)$ grows unboundedly as $n \to \infty$ .

The study is situated within the context of GAGA-type theorems for complex manifolds equipped with holomorphic contractions. A central tension exists between algebraic and holomorphic properties: while the Poincaré–Dulac theorem guarantees that certain contractions are holomorphically linearizable, it is not immediately clear if they are algebraically linearizable. The property of having bounded degree growth is an algebraic conjugation invariant; thus, if a map has unbounded degree growth, it cannot be algebraically conjugate to a linear map (which has constant degree).

Methodology and Definitions
The author employs a combination of algebraic group theory and dynamical systems analysis:

Strict Algebraicity: The paper introduces the concept of a "strictly algebraic" automorphism. An automorphism $\gamma$ is strictly algebraic if it belongs to an algebraic group action. Proposition 1 establishes that $\gamma$ is strictly algebraic if and only if its degree growth is bounded ( $\sup_n \deg \gamma^n < +\infty$ ).
Dimensional Analysis ( $d=2$ ): For the two-dimensional case, the author relies on the classification of automorphisms of $\mathbb{C}^2$ $C^{2}$ (Milnor, Friedland) and entropy results (Smillie). The argument proceeds by elimination:
- Affine and elementary automorphisms have constant degree growth.
- Loxodromic automorphisms (finite compositions of Hénon maps) exhibit exponential degree growth but possess positive topological entropy.
- Since a contraction must have zero topological entropy (no periodic points other than the origin), it cannot be loxodromic.
- Conclusion for $d=2$ : No polynomial contraction with unbounded degree growth exists.
Construction for $d \ge 3$ : The author constructs a specific affine-triangular automorphism on $\mathbb{C}^3$ $C^{3}$ (and by extension $\mathbb{C}^d$ $C^{d}$ ) defined by:
$\gamma(x, y, z) = (\lambda_1(y + xz), \lambda_2 x, \lambda_3 z)$
where $0 < \lambda_i < 1/2$ $0 < λ_{i} < 1/2$ .
- Contraction Proof: By analyzing the recursive relations of the iterates, the author demonstrates that the sequence converges to zero, satisfying the contraction definition.
- Degree Growth Calculation: By tracking the degrees of the component polynomials $F_n, G_n, H_n$ of $\gamma^n$ , the author derives the recurrence $\deg \gamma_{n+1} = \deg \gamma_n + 1$ , resulting in linear degree growth ( $\deg \gamma_n = n+1$ ).

Key Results

Non-existence in Dimension 2: Any contraction automorphism of $\mathbb{C}^2$ has bounded degree growth.
Existence in Dimension 3 and Higher: For any $d \ge 3$ , there exist polynomial contractions with unbounded (specifically linear) degree growth.
Holomorphic vs. Algebraic Linearization:
- By choosing parameters $\lambda_i$ to be algebraically independent, the eigenvalues of the differential at the origin avoid resonant relations.
- According to the Poincaré–Dulac theorem, such a map is holomorphically conjugate to a linear map.
- However, because the degree growth is unbounded, the map is not strictly algebraic and therefore not algebraically conjugate to any linear map.

Significance and Claims
The paper claims to provide a simple, explicit counterexample demonstrating a divergence between holomorphic and algebraic linearization for polynomial contractions in dimensions $d \ge 3$ .

Algebraic vs. Holomorphic: The constructed automorphisms are holomorphically linearizable (conjugate to a linear map via a holomorphic diffeomorphism) but not algebraically linearizable (not conjugate to a linear map via a polynomial automorphism).
Answer to Verbitsky-Ornea: The work answers the specific question regarding the existence of polynomial contractions with unbounded degree growth in the affirmative for $d \ge 3$ and the negative for $d=2$ .
Scope: The results are presented as a direct answer to the posed question, utilizing standard tools from the theory of algebraic groups and complex dynamics without proposing further applications or experimental extensions. The author notes that the construction extends to higher dimensions by scaling additional coordinates.

Technical Summary: Polynomial Contractions of Cd\mathbb{C}^dCd and Degree Growth

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Technical Summary: Polynomial Contractions of $\mathbb{C}^d$ and Degree Growth