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

Dmitrii Korshunov

arxiv: 2605.29386 · v1 · pith:QBN35ZGTnew · submitted 2026-05-28 · 🧮 math.CV · math.AG· math.DS

Polynomial contractions of mathbb C^d and degree growth

Dmitrii Korshunov This is my paper

Pith reviewed 2026-06-29 00:10 UTC · model grok-4.3

classification 🧮 math.CV math.AGmath.DS

keywords polynomial automorphismsdegree growthcontractionslinearizabilityPoincaré-Dulac theoremautomorphisms of C^dalgebraic maps

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The pith

There exists a polynomial contraction automorphism of C^d for d at least 3 whose iterates have unbounded degree growth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs an explicit polynomial automorphism of complex d-space in dimensions three and higher that contracts all orbits to a fixed point. The iterates of this map nevertheless have degrees that increase without bound under composition. Combined with the Poincaré-Dulac theorem on local linearization, the example produces an algebraic automorphism of C^d that admits a holomorphic linearizing change of coordinates but no algebraic one. A reader following the argument sees a concrete separation between holomorphic and algebraic linearizability within the class of polynomial maps.

Core claim

There exists a polynomial contraction automorphism of C^d, d≥3, with unbounded degree growth. Combined with the Poincaré-Dulac theorem this yields an algebraic automorphism of C^d, d≥3, which is holomorphically but not algebraically linearizable.

What carries the argument

An explicit polynomial map on C^d that is a contraction to the origin yet whose successive iterates have strictly increasing total degree.

Load-bearing premise

The explicit construction of a polynomial map that is simultaneously a contraction and has iterates of strictly increasing degree.

What would settle it

Direct verification that the given polynomial map either fails to contract all orbits to a fixed point or that the degrees of its iterates remain bounded would disprove the claim.

read the original abstract

We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincar\'e-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which is holomorphically but not algebraically linearizable.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a simple explicit polynomial automorphism of C^d (d≥3) that is a contraction with strictly increasing iterate degrees, yielding a holomorphic-but-not-algebraic linearizability distinction via Poincaré-Dulac, but the abstract gives no map or verification.

read the letter

The central claim is that there exists a polynomial automorphism f of C^d (d≥3) with 0 as a global attractor, spectral radius of Df(0) less than 1, and deg(f^n) going to infinity. This is then fed into Poincaré-Dulac to produce an algebraic automorphism that is holomorphically linearizable at 0 but cannot be algebraically linearizable, since a polynomial conjugacy would bound the degrees.

The paper does one thing cleanly: it isolates the degree-growth obstruction as the precise feature that separates the two linearizability notions in dimension three and higher. That framing is direct and matches known facts about algebraic versus holomorphic conjugacies.

The soft spot is obvious and load-bearing. The abstract asserts the existence of such an f but supplies neither the polynomial itself nor any calculation showing it is bijective with polynomial inverse, that it contracts, or that the degrees actually increase. Polynomial automorphisms in several variables are constrained; global invertibility is not automatic. Without the explicit map and the checks, the claim reduces to an existence statement whose verification is left to the reader.

If the full text contains a concrete polynomial together with the degree and invertibility arguments, the example is worth having. If it does not, the paper is just a sketch. Either way, the idea itself is not circular and engages the right literature.

This is for people working on holomorphic dynamics or algebraic automorphisms who need concrete examples. A reader who wants to test general linearization theorems against an explicit map would get value from it once the construction is checked. It deserves a serious referee to look at the map and the degree calculations.

Referee Report

2 major / 0 minor

Summary. The manuscript asserts the existence of a simple polynomial contraction automorphism of C^d (d≥3) with unbounded degree growth of its iterates. Combined with the Poincaré-Dulac theorem, this is claimed to yield an algebraic automorphism of C^d that is holomorphically linearizable at the origin but not algebraically linearizable.

Significance. If the asserted construction can be verified, the result would supply a concrete example separating holomorphic and algebraic linearizability for polynomial automorphisms of affine space in dimension ≥3, which is of interest to complex dynamics and the study of automorphism groups.

major comments (2)

[Abstract] Abstract: the central claim rests on an explicit construction of a polynomial automorphism f that is simultaneously a global contraction (0 globally attracting with spectral radius of Df(0)<1) and satisfies deg(f^n)→∞, yet no such map, no degree computations for the iterates, and no argument establishing bijectivity with polynomial inverse are supplied.
[Abstract] Abstract: the application of Poincaré-Dulac to obtain holomorphic linearizability is invoked without any local normal-form calculation or verification that the linear part satisfies the necessary non-resonance or resonance conditions for the theorem to apply in this setting.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the identification of points where the presentation requires greater explicitness. We address the two major comments below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim rests on an explicit construction of a polynomial automorphism f that is simultaneously a global contraction (0 globally attracting with spectral radius of Df(0)<1) and satisfies deg(f^n)→∞, yet no such map, no degree computations for the iterates, and no argument establishing bijectivity with polynomial inverse are supplied.

Authors: We agree that the abstract does not contain these elements. The body of the manuscript states the existence of such an f but does not display the explicit polynomial expressions, the inductive degree calculations, or the inverse map. We will revise the manuscript by inserting the concrete polynomial map (in coordinates), the recursive formulae or matrix representations used to track deg(f^n), the verification that the spectral radius condition holds and that the origin is globally attracting, and the explicit polynomial inverse that establishes bijectivity. These additions will appear in a new dedicated section. revision: yes
Referee: [Abstract] Abstract: the application of Poincaré-Dulac to obtain holomorphic linearizability is invoked without any local normal-form calculation or verification that the linear part satisfies the necessary non-resonance or resonance conditions for the theorem to apply in this setting.

Authors: We acknowledge that the manuscript invokes the Poincaré-Dulac theorem without performing the local analysis. In the revision we will add a short subsection that computes the linear part at the origin, checks the relevant resonance relations among its eigenvalues, and carries out the necessary normal-form computation to confirm that the theorem applies and yields holomorphic linearizability. The algebraic non-linearizability will remain unchanged. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction supplies the claimed example

full rationale

The paper's central claim is an existence statement resting on an explicit polynomial automorphism of C^d (d≥3) that is simultaneously a global contraction and has strictly increasing iterate degrees. No equations, fitted parameters, or self-referential definitions appear; the Poincaré-Dulac application is a standard external theorem, and the algebraic-vs-holomorphic distinction follows directly from the degree-growth property of the constructed map. The derivation is therefore self-contained and does not reduce any prediction or uniqueness claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information from abstract alone; no free parameters, axioms, or invented entities can be identified.

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discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Coherent sheaves on subvarieties in Hopf manifolds
math.AG 2026-06 unverdicted novelty 6.0

Establishes GAGA for varieties with contractions, a quotient structure for Hopf subvarieties, and filtrability of reflexive coherent sheaves on them.

Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages · cited by 1 Pith paper · 1 internal anchor

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work page internal anchor Pith review Pith/arXiv arXiv 2018
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Dynamical degrees of affine-triangular auto- morphisms of affine spaces.Ergodic Theory and Dynamical Systems, 42(12):3551–3592, 2022

J´ er´ emy Blanc and Immanuel Van Santen. Dynamical degrees of affine-triangular auto- morphisms of affine spaces.Ergodic Theory and Dynamical Systems, 42(12):3551–3592, 2022

2022

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Springer Science & Business Media, 2012

Armand Borel.Linear algebraic groups. Springer Science & Business Media, 2012

2012

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Holomorphically conjugate polynomial auto- morphisms ofC 2 are polynomially conjugate.Bulletin of the London Mathematical Society, 56(12):3745–3751, 2024

Serge Cantat and Romain Dujardin. Holomorphically conjugate polynomial auto- morphisms ofC 2 are polynomially conjugate.Bulletin of the London Mathematical Society, 56(12):3745–3751, 2024

2024

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Dynamical properties of plane polynomial auto- morphisms.Ergodic Theory and Dynamical Systems, 9(1):67–99, 1989

Shmuel Friedland and John Milnor. Dynamical properties of plane polynomial auto- morphisms.Ergodic Theory and Dynamical Systems, 9(1):67–99, 1989

1989

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On the geometry of the automorphism groups of affine varieties

Jean-Philippe Furter and Hanspeter Kraft. On the geometry of the automorphism groups of affine varieties.arXiv preprint arXiv:1809.04175, 2018

work page internal anchor Pith review Pith/arXiv arXiv 2018

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Springer, 2024

Liviu Ornea and Misha Verbitsky.Principles of locally conformally K¨ ahler geometry. Springer, 2024

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