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arxiv: 2604.21398 · v2 · submitted 2026-04-23 · 🧮 math.AG · math.DS· math.RA

A lower bound for polynomial volume growth of automorphisms of zero entropy

Fei Hu , Chen Jiang This is my paper

Pith reviewed 2026-05-12 02:53 UTC · model grok-4.3

classification 🧮 math.AG math.DSmath.RA
keywords zero-entropy automorphismspolynomial volume growthprojective varietiesdynamical intersection polynomialsGelfand-Kirillov dimensionintersection numbersgap principlealgebraic dynamics
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The pith

Zero-entropy automorphisms of normal projective varieties satisfy plov(f) ≥ d + k(k+2)/4.

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

The authors aim to establish a sharp lower bound for the polynomial volume growth of zero-entropy automorphisms on normal projective varieties of dimension d. With the first-degree growth rate k defined by the asymptotic deg_1(f^n) ~ n^k, they show plov(f) is at least d plus k(k+2)/4. This matters because it provides the best possible estimate and equivalently bounds the Gelfand-Kirillov dimension of the twisted homogeneous coordinate ring from below. They develop dynamical intersection polynomials to give a new characterization of plov(f) using non-vanishing intersection numbers. The work also proves a gap principle for the possible values of plov(f) when d is at least 4.

Core claim

For a normal projective variety X of dimension d and zero-entropy automorphism f with deg_1(f^n) asymptotic to n^k, the polynomial volume growth plov(f) is at least d + k(k+2)/4. This is sharp and gives a sharp lower bound on the Gelfand-Kirillov dimension of the twisted homogeneous coordinate ring. Dynamical intersection polynomials are introduced to characterize plov(f) by the non-vanishing of intersection numbers. A gap principle holds: for d ≥ 4 either plov(f) = d^2 or plov(f) ≤ d(d-2) + 2 floor(d/4). In dimension 4 all possible values are determined.

What carries the argument

Dynamical intersection polynomials that characterize the polynomial volume growth through non-vanishing of intersection numbers.

Load-bearing premise

The first-degree growth rate k exists as the exponent where deg_1(f^n) grows like n to the power k, and the dynamical intersection polynomials satisfy the required positivity and non-vanishing conditions on the normal projective variety.

What would settle it

An explicit zero-entropy automorphism f on a projective variety of dimension d where the polynomial volume growth plov(f) is calculated to be smaller than d + k(k+2)/4 would falsify the main lower bound.

read the original abstract

Let $X$ be a normal projective variety of dimension $d$, and let $f$ be a zero-entropy automorphism of $X$. Denote by $k$ the first-degree growth rate of $f$, so that $\deg_1(f^n) \asymp n^{k}$. We prove the sharp lower bound for the polynomial volume growth $\mathrm{plov}(f)$ of $f$: \[ \mathrm{plov}(f) \ge d+\frac{k(k+2)}{4}, \] equivalently giving a sharp lower bound on the Gelfand--Kirillov dimension of the associated twisted homogeneous coordinate ring. This improves previous lower bounds of Keeler and of Lin--Oguiso--Zhang. In the proof, we introduce the notion of dynamical intersection polynomials and give a new characterization of $\mathrm{plov}(f)$ in terms of non-vanishing of intersection numbers. We also establish a gap principle for polynomial volume growth: for every fixed dimension $d\ge 4$, either $\mathrm{plov}(f)=d^2$, or $\mathrm{plov}(f)\le d(d-2) + 2\lfloor d/4 \rfloor$. This reveals a new rigidity phenomenon for zero-entropy automorphisms. As an application, in dimension $4$ we determine all possible values of $\mathrm{plov}$, thereby extending the results of Artin--Van den Bergh for surfaces and Lin--Oguiso--Zhang for threefolds.

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.

Referee Report

0 major / 4 minor

Summary. The paper proves a sharp lower bound plov(f) ≥ d + k(k+2)/4 for the polynomial volume growth of a zero-entropy automorphism f of a normal projective variety X of dimension d, where k is the first-degree growth rate with deg_1(f^n) ≍ n^k. It introduces dynamical intersection polynomials to give a new characterization of plov(f) in terms of non-vanishing intersection numbers, establishes a gap principle (for d ≥ 4, either plov(f) = d² or plov(f) ≤ d(d-2) + 2⌊d/4⌋), and classifies all possible values of plov in dimension 4, improving on prior bounds of Keeler and Lin-Oguiso-Zhang. The result is equivalently phrased as a lower bound on the Gelfand-Kirillov dimension of the associated twisted homogeneous coordinate ring.

Significance. If the central claims hold, the work supplies a sharp improvement on existing lower bounds and introduces dynamical intersection polynomials as a new tool extending standard intersection theory on projective varieties. The gap principle reveals a previously unobserved rigidity phenomenon for zero-entropy automorphisms, while the dimension-4 classification extends known results for surfaces and threefolds. The equivalence to Gelfand-Kirillov dimension provides a clean bridge to noncommutative algebra. These contributions are likely to influence future work on algebraic dynamics and growth rates.

minor comments (4)
  1. [Abstract and §1] The asymptotic notation ≍ in the abstract and §1 should be defined explicitly (or a reference to a standard source given) to avoid any ambiguity for readers outside algebraic dynamics.
  2. [Theorem on gap principle] In the statement of the gap principle (Theorem 1.3 or equivalent), an explicit low-dimensional example (e.g., for d=4) would help illustrate the two possible regimes and strengthen the claim of a new rigidity phenomenon.
  3. [Introduction] The comparison with previous bounds of Keeler and Lin-Oguiso-Zhang in the introduction would benefit from citing the precise statements of their theorems rather than only naming the authors.
  4. [§3] Notation for the dynamical intersection polynomials (introduced in §3) is clear once defined, but a short table summarizing their key positivity and growth properties would improve readability when they are used to characterize plov(f).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work, the assessment of its significance, and the recommendation for minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected in the derivation

full rationale

The paper introduces dynamical intersection polynomials as an extension of standard intersection theory on normal projective varieties. It provides a new characterization of plov(f) via non-vanishing of intersection numbers and derives the lower bound plov(f) ≥ d + k(k+2)/4 from positivity and growth properties under the zero-entropy assumption. k is independently defined from the first-degree growth rate deg_1(f^n) ≍ n^k. No step reduces by definition or construction to the target bound, no fitted inputs are renamed as predictions, and no load-bearing self-citations or ansatzes are invoked. The gap principle and Gelfand-Kirillov equivalence follow as direct consequences. The argument is self-contained against external benchmarks from intersection theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed from abstract alone; ledger therefore records only items explicitly named in the abstract.

axioms (1)
  • standard math Standard facts from algebraic geometry on normal projective varieties, intersection theory, and automorphisms
    Invoked implicitly when defining deg_1, plov, and zero entropy.
invented entities (1)
  • dynamical intersection polynomials no independent evidence
    purpose: New characterization of plov(f) via non-vanishing of intersection numbers
    Introduced in the paper to prove the lower bound and gap principle.

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