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For any degree d≥2, non-periodic Hénon points have canonical height bounded away from zero.

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2026-08-02 06:25 UTC pith:G5RE4GOU

load-bearing objection Plausibly useful extension of Ingram's d=2 bound to all degrees, but the final discreteness step and the period-B condition need fixing before the theorem is proved. the 5 major comments →

arxiv 2607.12668 v3 pith:G5RE4GOU submitted 2026-07-14 math.NT

On lower bounds for canonical heights of the map φ(X,Y)=(Y,X+Y^D+b)

Jorge Mello This is my paper
classification math.NT MSC 11G5037P30
keywords Hénon mapcanonical heightlower boundarithmetic dynamicsnon-periodic pointslocal heightspigeonhole principlefunction fields
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to prove a uniform lower bound for the canonical height of non-periodic points under the plane automorphisms φ(x,y)=(y,x+y^d+b) over number and function fields of characteristic 0. For every d≥2 there is a finite period bound B such that every point whose orbit is not periodic of period at most B has canonical height at least ε·max{h(b),1}, where ε depends only on d and the places dividing b. This generalizes the previously known quadratic case to all degrees and gives evidence for classical conjectures predicting that canonical heights cannot be arbitrarily small outside periodic points. A corollary limits the coincidences of orbits of two distinct points at s-integral values of a parameter t.

Core claim

The central claim is that the positive values of the canonical height ĥ_φ on A²(K) are bounded below by a constant multiple of max{h(b),1}, up to excluding points periodic with period at most a finite B. The proof derives a global comparison h(b) ≤ d^{M+2} ĥ_φ(P) + C for any non-periodic P, using local height estimates at all places; combined with the discreteness of the canonical height, this forces the claimed lower bound. The same inequality yields the finiteness statement for coincident orbits in the function-field setting.

What carries the argument

The main mechanism is the decomposition of the global canonical height into local heights, with explicit 'escaping' regions B±_v where the local height equals log|x| or log|y| up to a bounded error. Lemmas 2.3–2.6 use pigeonhole arguments across an interval of iterates to find many indices whose coordinates are v-adically close, bounding their distances by d^M ĥ_φ(P) plus constants. At primes of bad reduction, approximations by the d-th roots of −b control the differences. Summing over all places converts these local estimates into the global inequality that drives the lower bound.

Load-bearing premise

The proof assumes that the set of nonzero canonical heights among points in K² has a positive minimum, citing discreteness in the literature; discreteness of a set of nonnegative real numbers does not by itself rule out 0 as an accumulation point, so this step—not the local estimates—carries the final inequality.

What would settle it

Exhibit, for some d>2 and some base field K, a non-periodic point P with canonical height satisfying 0 < ĥ_φ(P) < ε max{h(b),1} for the constants ε of Theorem 1.1; or construct an infinite sequence of non-periodic points with h(b) bounded and ĥ_φ(P) → 0. Either would directly refute the claimed uniform lower bound and, more fundamentally, would show that the positive canonical heights do not have a positive minimum.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If the theorem is correct, the canonical-height spectrum of these Hénon maps has a gap above zero: non-periodic points cannot have arbitrarily small positive canonical height when h(b) is fixed.
  • It supplies the first general lower-bound result of this shape for Hénon maps of degree greater than two, extending the quadratic case.
  • Over function fields, it yields that for two points with distinct orbits, the set of s-integral parameters t for which the orbits coincide is finite.
  • The bound is uniform in the point P, depending only on the map's degree and the places dividing b.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The theorem likely extends to maps (x,y) ↦ (ay, x+f(y)) with arbitrary leading coefficient a and general polynomial f of degree d, since the local-height framework is built for that form; the paper only states the monic normalized case.
  • The proof's B is defined recursively (B_{j+1,n}=(d^4+2)B_{j,n}+d^4+2), so the excluded periodic periods are explicit but almost certainly not optimal; computing the true maximal gap in the height spectrum could produce much smaller B.
  • A natural testable extension is to make ε explicit in terms of d and the primes dividing b, then search for points with canonical height below the bound; any such point would pinpoint where the argument fails.
  • The method appears to depend only on the height doubling along the escaping regions, not on the specific form y^d, so a similar lower bound may hold for any 'dominant' term of degree d in the second coordinate.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper claims a lower bound for the canonical height ĥ_φ of non-periodic points of Hénon maps φ(x,y)=(y, x+y^d+b), d≥2, over number fields or function fields of characteristic 0: there is a constant ε>0 and an integer B such that any point not periodic of period at most B satisfies ĥ_φ(P) ≥ ε max{h(b),1}. The proof follows Ingram's strategy for d=2: local canonical heights, pigeonhole extraction of two iterates that are close at all places, and a global height inequality h(b) ≤ d^{M+2}ĥ_φ(P)+C, which is then converted into a lower bound by citing discreteness of the canonical height. The paper also derives a corollary on unlikely intersections over function fields.

Significance. If the theorem is correct, it gives the first Ingram-type lower canonical-height bound for Hénon maps of degree d>2 and supports conjectures of Lang-Silverman flavor. The strategy is natural and largely follows the established Ingram framework; the local estimates are explicit and do not appear to be fitted to the conclusion. However, the proof as written has several load-bearing gaps: the use of 'discreteness' to obtain a positive minimum is not justified as stated, the theorem's hypothesis about period at most B is logically inadequate, and Lemma 2.3 contains an additive/logarithmic inconsistency that breaks the global summation. These issues are substantial but appear repairable if [2] supplies the necessary Northcott-type finiteness and the statement/proof are corrected.

major comments (5)
  1. [Section 3, final paragraph] The step 'the fact that ĥ_φ is discrete [2] yields h(b)≤d^{M+2}·min{ĥ_φ(P)|P∈K², ĥ_φ(P)≠0}+C' is not justified. Discreteness of a subset of [0,∞) does not preclude 0 being an accumulation point, e.g. {1/n}. The proof needs a Northcott-type finiteness statement for bounded height intervals, or an explicit positive lower bound for all nonzero ĥ_φ(P). This is the only place where the '1' in max{h(b),1} is obtained when h(b) is small. Please state precisely what [2] implies and, since the theorem covers function fields, verify that the required finiteness holds there.
  2. [Theorem 1.1 and Section 3] The hypothesis 'not periodic for φ of period at most B' is weaker than 'not periodic'. A point of period >B satisfies the hypothesis but has ĥ_φ(P)=0, directly contradicting the conclusion unless B is a uniform bound for all periods. The proof never defines B; the recursive constants B_{m,n} are used only to choose M, and no relation between B_{m,n} and the theorem's B is given. Please either state the theorem for all nonperiodic points and prove/cite a uniform period bound, or explicitly exclude all periodic points.
  3. [Lemma 2.3 and its use in Section 3] The lemma's displayed inequality is additive: |x_i−x_j|_v + |y_i−y_j|_v + λ_v(b) ≤ 3d^M ĥ_{v,φ}(P)+α_v. The proof, however, proves a logarithmic inequality, e.g. Eq. (1): log|x_i−x_j|_v + log|y_i−y_j|_v + λ_v(b) ≤ ... . Section 3 then sums over places to obtain h(b), which requires logarithmic heights. As stated, the additive lemma cannot produce the global height bound used in the main proof. The lemma statement and proof must be made consistent, and the convention 'if x_i=x_j' should be handled for the logarithmic form.
  4. [Section 3, subset construction and B definition] The sentence 'By Lemma 2.3 applied to places of bad reduction and Lemma 2.6 applied to places of good reduction, we can choose a subset I⊂[-M,M] with #I≥B_{0,s}' is only a sketch. The iterative pigeonhole that simultaneously preserves the desired inequality for all places must be written out; the constants B_{m,n} appear to encode this induction, but the argument is not given. Moreover, the theorem's B is never identified with B_{s,s} or otherwise defined. Without this induction, the existence of I and the final constant C are not established.
  5. [Scope: number field vs function field] Section 2 begins 'defined over a number field K', but Theorem 1.1 and Corollary 1.2 are stated over number fields or function fields. The proof does not indicate which lemmas—especially the global decomposition of ĥ_φ and the discreteness/Northcott property—remain valid over function fields. If the function-field case is intended, the missing justification is essential; if not, the theorem's statement should be restricted to number fields.
minor comments (5)
  1. [Abstract] The abstract says the lower bound is for 'D>2' and 'extends previous work for D=2', but Theorem 1.1 states d≥2. Please make the range of d consistent.
  2. [Introduction] There is a typo: 'we can can give more evidence' should be 'we can give more evidence'.
  3. [Lemma 2.2] The notation 'there are roots γ_1^d=-b and γ_2^d=-b' is confusing; the two roots are not otherwise distinguished. Clarify the statement.
  4. [Lemma 2.6] The proof concludes |x_i−x_j|_v ≤ ..., while the lemma states log|x_i−x_j|_v ≤ ... . Align the statement and proof.
  5. [Section 3] The sentence 'If the latter was true, we use Lemma 2.4 instead of Lemma 2.5 to reach the same conclusion' is a leftover and should be removed or rewritten.

Circularity Check

0 steps flagged

No significant circularity: the proof derives a new lower bound from prior independent results by Ingram and Kawaguchi; no fitted constant is renamed as a prediction and no load-bearing self-citation appears.

full rationale

The paper's central derivation is not circular. It starts from the definitions of canonical heights and local heights, proves explicit geometric lemmas (Lemmas 2.1–2.6), and then follows the structure of Ingram's argument [1] to obtain the inequality h(b) ≤ d^{M+2} ĥ_φ(P) + C. The final use of Kawaguchi's discreteness result [2] is an external, prior result with its own stated assumptions; it is not a restatement of the theorem being proved, nor is it authored by the present paper's author. Even if one doubts whether 'discrete' alone justifies a positive minimum over nonzero heights, that is a correctness or support gap, not a circularity: the target lower bound is not assumed or fit into the argument. No parameter is fitted to the conclusion, and the theorem's ε is an explicit constant arising from the proof, not a named prediction equal to an input. The paper also does not rename a known result: it genuinely extends Ingram's d = 2 bound to d ≥ 2. Therefore no circular step is present.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters fitted; the paper is a pure proof. Main unstated inputs are the product formula, the existence of a positive minimum for nonzero canonical heights, and the finiteness of periodic points (to justify B).

axioms (4)
  • domain assumption Global canonical heights equal sums of local canonical heights over all places (Kawaguchi).
    Section 2, first paragraph; used in every summation over places in Section 3.
  • standard math Product formula: for a∈K^*, sum_v [K_v:Q_v]/[K:Q] log|a|_v = 0.
    Used implicitly when summing Lemma 2.3's log inequalities over places and dropping log|xi-xj|+log|yi-yj|; not stated in Section 3.
  • domain assumption The set of positive canonical heights on A^2(K) has a positive minimum (or Northcott-style finiteness suffices).
    Section 3, final paragraph, where 'ĥ_φ is discrete [2]' is used to get a lower bound for small h(b); discreteness alone does not imply a positive minimum.
  • domain assumption Periodic points in A^2(K) have uniformly bounded (finite) periods, so B can be chosen larger than 2M and all periods.
    Implicit in the statement of Theorem 1.1 and in excluding P_i=P_j for i≠j in I; relies on finiteness of periodic points (Silverman [6]) but the proof never says B is chosen this way.

pith-pipeline@v1.3.0-alltime-deepseek · 9138 in / 33475 out tokens · 312819 ms · 2026-08-02T06:25:13.668843+00:00 · methodology

0 comments
read the original abstract

We give a lower bound for the canonical height associated to H\'enon maps $\phi(X,Y)=(Y,X+Y^D+B)$ of non-periodic points when $D>2$ in the spirit of conjectures of Lang and Silverman. This is followed by an application and extends previous work for $D=2$ by Ingram.

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