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Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties

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Pith's one-line read This paper proves that along a generic orbit of a dominant rational self-map of a smooth projective variety, the height measuring generalized greatest common divisors among orbit points is asymptotically negligible compared with any ample…

desk verdict New and substantial result for rational maps, but the morphism-case reduction for arbitrary Y is flawed because the height inequality h_Y ≤ h_Z is backwards. read the letter →

arxiv 2507.05027 v1 pith:IEZIOBAQ submitted 2025-07-07 math.AG math.DSmath.NT

Yohsuke Matsuzawa This is my paper · ORCID
classification math.AGmath.DSmath.NT MSC 37P1537P55
keywords ArithmeticdynamicsGeneralizedgreatestcommondivisorsHeightassociatedwithsubschemesdegreeDynamicalRationalself-mapsProjectivevarieties
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves, without assuming the standard Diophantine conjecture used in earlier work, that the height attached to a closed subscheme of codimension $c$ — the 'generalized greatest common divisor' of the orbit points — is asymptotically negligible along a generic orbit of a dominant rational self-map. Concretely, Theorem 1.5 shows $\lim_{n\to\infty} h_Y(f^n(x))/h_H(f^n(x)) = 0$ whenever the orbit of $x$ is generic, the subscheme $Y$ is suitably transverse (or $f$ is a morphism), and the $c$-th dynamical degree satisfies $d_c(f)^{1/c} < \alpha_f(x)$, the arithmetic degree of $x$. Earlier results on this problem were conditional on the standard Diophantine conjecture; the present proof uses the expansion of the dynamics itself and therefore applies only when the dynamical degrees are not all equal. The conclusion is the expected one: the number of digits of common divisors among orbit coordinates is smaller than the number of digits of the coordinates themselves.

What carries the argument

One mechanism carries the proof: converting a nonzero section of the ideal-sheaf power $I^r(mH)$ into an effective divisor $D \sim mH$ that contains the $r$-th thickening of the subscheme, which yields $r h_Z \leq m h_H + O(1)$ away from $D$. The work is to show such sections exist with $m/r$ as small as $(d_{N-l}(f)^{1/(N-l)} + \varepsilon)^n$ after replacing $Z$ by $f^{-n}(Y)$. This is done in Proposition 2.10 by bounding the Hilbert function of the thickened $f^{-n}(Y)$ via general hyperplane sections: because $Y$ sits inside the finite-iteration locus and is regularly embedded, the Segre class of $f^{-n}(Y)$ is controlled by the normal bundle, giving $\tau(f^{-n}(Y) \cap V, V) \leq C (f^n)^* H^{N-i} \cdot H^i$ for general hyperplanes. The resulting bound $h_{f^{-n}(Y)} \leq (d_{N-l}(f)^{1/(N-l)} + \varepsilon)^n h_H + O(1)$ outside a proper closed subset is then matched against a lower bound on $h_H$ along the orbit whose exponential rate is arbitrarily close to $\alpha_f(x)$, forcing the ratio to zero.

What would settle it

Find a dominant rational map $f$, a closed subscheme $Y$ satisfying the theorem's geometric hypotheses, and a point $x$ with generic orbit and $d_{N-l}(f)^{1/(N-l)} < \alpha_f(x)$ for which the limsup of $h_Y(f^n(x))/h_H(f^n(x))$ is positive; all quantities are explicit for monomial maps on $\mathbb{G}_m^N$, where one can compute both heights and check whether the ratio tends to zero.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.5. For a smooth projective variety $X$ over $\overline{\mathbb{Q}}$ of dimension $N$, a dominant rational self-map $f$, and a proper closed subscheme $Y$ of dimension $l$ with $c = N - l$, the theorem says that if the orbit of $x$ is well-defined and generic, and if $d_c(f)^{1/c} < \alpha_f(x)$, then $h_Y(f^n(x))/h_H(f^n(x))$ tends to $0$ for every ample height $h_H$. When $f$ is not a morphism, $Y$ must be pure dimensional, a regular embedding in $X$, and contained in the finite-iteration locus $X_f^{\mathrm{back}}$; for morphisms the theorem shows these extra conditions can be arranged by replacing $Y$ with a general complete intersection containing it. The paper also exhibits a counterexample showing that dropping the finite-iteration condition breaks the conclusion, with the ratio tending to $1$ instead of $0$.

Load-bearing premise

The final step leans on a previously established lower bound asserting that along the orbit the height grows at least like $(\eta \alpha_f(x))^{mk}$ with $\eta$ arbitrarily close to $1$; if that bound fails for a point that otherwise meets Theorem 1.5's hypotheses, the proof no longer goes through.

Editorial extensions

If this is right

  • For morphisms the theorem applies to every proper closed subscheme $Y$: after replacing $Y$ by a general complete intersection containing it, the ratio $h_Y(f^n(x))/h_H(f^n(x))$ tends to $0$ for every generic orbit with $d_{N-l}(f)^{1/(N-l)} < \alpha_f(x)$.
  • If the standard conjectures hold that Zariski-dense orbits are generic and that their arithmetic degree equals the first dynamical degree, the hypotheses become simply: the orbit is Zariski dense and $d_{N-l}(f)^{1/(N-l)} < d_1(f)$.
  • The finite-iteration hypothesis on $Y$ is essential: for $f(x:y:z)=(x^2y:y^3:z^3)$ on $\mathbb{P}^2$ and $Y=\{0:0:1\}$, a point whose orbit is generic and satisfies the degree inequality has $h_Y(f^n(x))/h_H(f^n(x)) \to 1$.
  • For monomial maps on the multiplicative torus induced by an integer matrix with $|\lambda_1| > |\lambda_N|$, any Zariski-dense orbit with a zero-dimensional $Y$ satisfies the ratio limit, since $d_N(f)^{1/N} = |\lambda_1\cdots\lambda_N|^{1/N} < |\lambda_1| = d_1(f)$.
  • For maps with $d_1(f) > d_2(f)$, a point with arithmetic degree equal to $d_1(f)$ automatically has Zariski-dense orbit; combined with the standard genericity conjecture for such orbits, the main theorem applies.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof reveals a controlling principle the paper does not state as such: the $c$-th dynamical degree, not the first, governs how much 'gcd information' a codimension-$c$ subscheme can accumulate along an orbit; this suggests that for maps with $d_c^{1/c}=d_1$ (such as polarized endomorphisms), a different invariant—likely ramification along $Y$—would be needed to get any vanishing at all.
  • Because the main estimate is an upper bound with explicit exponential rate, a quantitative version giving a decay rate for $h_Y(f^n(x))/h_H(f^n(x))$ should follow from the same machinery; the paper does not write one down.
  • The morphism-case reduction, which passes from an arbitrary $Y$ to a general complete intersection containing it, suggests a route to removing the regular-embedding hypothesis for rational maps as well, though the finite-iteration condition would still need to be handled.
  • The examples give a ready-made computational benchmark: for the monomial map examples one can compute both heights explicitly and verify the ratio's decay, which would serve as a numerical check of the theorem's scope.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper proves that for a dominant rational self-map f on a smooth projective variety X over \overline{\mathbb{Q}}, a proper closed subscheme Y, and a point x with generic well-defined orbit, the generalized gcd height h_Y(f^n(x)) is asymptotically negligible compared with any ample height h_H(f^n(x)), provided that a suitable dynamical degree is strictly smaller than the arithmetic degree of x. The main theorem (Theorem 1.5) gives this for a morphism with arbitrary Y, and for non-morphisms under the additional assumptions that Y is pure dimensional, regularly embedded, and contained in the iteratively finite locus X_f^{back}. The proof combines cohomological estimates for ideals of preimages f^{-n}(Y) with a lower bound for h_H along the orbit imported from the author's earlier work [24].

Significance. If the proof is completed, this is a strong unconditional result: it gives a dynamical criterion under which generalized greatest common divisors along orbits grow slower than every ample height, without assuming Vojta's conjecture. The technical heart, especially Proposition 2.10 and its Segre-class estimates, is substantial and appears to be new. The paper also gives useful examples showing the necessity of the condition Y\subset X_f^{back}. However, the morphism case of the main theorem is not proved as written because the reduction to pure-dimensional regularly embedded Y is based on a false height inequality. This must be repaired or the statement must be restricted.

major comments (1)
  1. [Section 2, proof of Theorem 1.5 (first paragraph, morphism-case reduction)] The reduction of the morphism case to the case where Y is pure dimensional and regularly embedded is invalid. The proof asserts that for a general sequence D_1,...,D_{N-l} in the linear system defined by H^0(X,I\otimes L), with Z=D_1\cap\cdots\cap D_{N-l}, one has h_Y\leq h_Z+O(1) because Y\subset Z. This inequality is false in general. For example, take X=\mathbb{P}^2, Y=(0:0:1), Z=\{x=0\}, and P_n=(2^n:2^n:1). Then h_Y(P_n)=n\log 2+O(1), while h_Z(P_n)=O(1) (in fact 0 for the standard height). Thus h_Y is not bounded above by h_Z. In general, for Y\subset Z the height inequality goes in the opposite direction: h_Z\leq h_Y+O(1). Consequently the morphism case of Theorem 1.5, which allows arbitrary proper closed subschemes Y, is not established by the given argument. The theorem either needs a different reduction (for instance, controlling h_Y through h_{Y_{\mathrm{red}}} and then handling the reduced scheme, if that is feasible) or its statement must be restricted to pure-dimensional regularly embedded Y in the morphism case as well.
minor comments (4)
  1. [Introduction, conventions] There is a duplicated phrase 'Namely, Namely' in the normalization paragraph; please fix it.
  2. [Lemma 2.9] The word 'infintie' should be 'infinite'. Similar typographical errors occur elsewhere, such as 'allow' for 'arrow' in Example 3.1.
  3. [Theorem 1.5 and abstract] The theorem states that Y has dimension l, while the abstract phrases the hypothesis as d_c(f)^{1/c}<\alpha_f(x) with c the codimension. For non-pure-dimensional Y the relation between l and c should be stated explicitly, or Y should be assumed pure dimensional in all cases.
  4. [Proof of Theorem 1.5, equation (2.12)] The final lower bound for h_H along the orbit is imported from [24, Lemma 2.4]. Since this is load-bearing, please state the lemma precisely and explain how [24, Theorem 2.2] identifies \alpha_f(x) with the relevant Lyapunov exponent so that the hypotheses of [24, Lemma 2.4] are satisfied for the point x in question.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main theorem is proved from independent cohomological estimates and prior non-circular self-citations; no hypothesis is a renamed version of the conclusion.

full rationale

The claimed derivation is not circular. Theorem 1.5 assumes a generic orbit and the inequality d_{N-l}(f)^{1/(N-l)} < α_f(x), and it proves h_Y(f^n(x))/h_H(f^n(x)) → 0. The proof splits into two independent estimates. Proposition 2.6 and Proposition 2.12 give an upper bound h_{f^{-n}(Y)}(x) ≤ (d_{N-l}(f)^{1/(N-l)}+ε)^n h_H(x)+O(1) using Hilbert-function bounds, Segre classes, and dynamical degrees; this is a new estimate and does not contain the target ratio. A lower bound h_H(f^{mk+s}(x)) ≥ C(η α_f(x))^{mk} h_H(f^s(x)) is imported from [24, Lemma 2.4]. That cited lemma is an independent published theorem with its own hypotheses; it does not assert anything about h_Y or about the ratio being zero, so the self-citation is not circular. The hypothesis d_{N-l}^{1/(N-l)} < α_f(x) is then used only to make the exponent in the upper bound smaller than the exponent in the lower bound; it is a sufficient condition, not an assumed conclusion. No parameter is fitted to the orbit data, and no quantity is defined in terms of the limit being proved. The external criticism that the morphism-case reduction uses the inequality h_Y ≤ h_{D_1∩...∩D_{N-l}}+O(1), which can fail for subschemes Z containing Y, is a genuine correctness concern about that reduction, but it is not circularity; it does not raise the circularity score. The residual self-citation in the proof is real but non-circular, so the score is 2 rather than 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim does not fit any data and does not introduce new physical or mathematical entities. It relies on standard background in height theory and dynamical degrees, plus two particular results from the author's prior article [24] used as black boxes. Auxiliary constants such as epsilon, delta, and eta are quantified bounds, not free parameters fitted to data.

assumptions (4)
  • standard math Heights associated with closed subschemes behave functorially: h_Y(f^n(x)) <= h_{f^{-n}(Y)}(x) + O(1), and h_Y is bounded below outside Y.
    Used throughout Sections 2 and 3; based on Silverman's theory [28] and standard height properties.
  • standard math Dynamical degrees converge and satisfy log-concavity for dominant rational self-maps.
    Used in Proposition 2.12 to compare d_{N-i}^{1/(N-i)} with d_{N-l}^{1/(N-l)}; cited from Dang, Truong, and Xie.
  • domain assumption For a generic orbit, the arithmetic degree exists and equals a cohomological Lyapunov exponent, and the lower bound (2.12) holds.
    Imported from Matsuzawa [24, Theorem 2.2 and Lemma 2.4]. This is the key external input that converts the dynamical degree inequality into the final ratio limit.
  • domain assumption In the morphism case, replacing Y by a general complete intersection Z = D_1 intersect ... intersect D_{N-l} containing Y preserves the needed height inequalities: h_Y <= h_Z + O(1).
    Used in the reduction step at the start of the proof of Theorem 1.5; the inequality is stated without a full proof.

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Cite this review

Pith. "Pith review of Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties." pith.science (2026). https://pith.science/paper/IEZIOBAQ

@misc{pith2026250705027,
  author       = {Pith},
  title        = {Pith review of: Growth of generalized greatest common divisors along orbits of self-rational maps on projective varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IEZIOBAQ}},
  note         = {Machine review of arXiv:2507.05027}
}
abstract

Consider a dominant rational self-map $f$ on a smooth projective variety $X$ defined over $\overline{\mathbb{Q}}$. We prove that \begin{align} \lim_{n \to \infty} \frac{h_{Y}(f^{n}(x))}{h_{H}(f^{n}(x)) } = 0, \end{align} where $h_{Y}$ is a height associated with a closed subscheme $Y \subset X$ of codimension $c$, $h_{H}$ is any ample height on $X$, and $x \in X(\overline{\mathbb{Q}})$ is a point with well-defined orbit, under the following assumptions: (1) either $f$ is a morphism, or $Y$ is pure dimensional, regularly embedded in $X$, and contained in the locus over which all iterates of $f$ are finite; (2) the orbit of $x$ is generic; (3) $d_{c}(f)^{1/c} < \alpha_{f}(x)$, where $d_{c}(f)$ is the $c$-th dynamical degree of $f$ and $ \alpha_{f}(x)$ is the arithmetic degree of $x$.

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