REVIEW 3 major objections 2 minor
Hénon maps of degree D>2 admit a positive lower bound on canonical height for every non-periodic point.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
T0 review · grok-4.5
2026-07-15 04:17 UTC pith:G5RE4GOU
load-bearing objection Abstract-only claim extending Ingram’s D=2 canonical-height lower bound to Hénon maps of every degree D>2; useful if true, but the estimates are unchecked. the 3 major comments →
On lower bounds for canonical heights of the map φ(X,Y)=(Y,X+Y^D+b)
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
There exists a positive lower bound for the canonical height ħ_φ of every non-periodic point of the Hénon map φ(X,Y)=(Y,X+Y^D+B) whenever D>2, extending the D=2 case.
What carries the argument
The canonical height ħ_φ attached to the Hénon map φ(X,Y)=(Y,X+Y^D+B); the argument shows this height stays strictly positive off the periodic points once D exceeds 2.
Load-bearing premise
The height estimates and comparison techniques that succeed for the quadratic case continue to produce a strictly positive lower bound for every integer degree greater than 2, without further restrictions on the constant B or the base field.
What would settle it
Exhibit a non-periodic rational or algebraic point of some map φ(X,Y)=(Y,X+Y^D+B) with D>2 whose canonical height is zero, or show that the height lower bound obtained by the paper’s estimates vanishes for some D>2.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims a positive lower bound on the canonical height ĥ_φ associated to the Hénon maps φ(X,Y)=(Y,X+Y^D+B) for every non-periodic point, in the regime D>2. This is presented as an extension of Ingram’s earlier result for the quadratic case D=2. The abstract alone supplies no explicit statement of the bound, its dependence on D, B or the base field, or any outline of the argument.
Significance. If the claimed uniform positivity of ĥ_φ for non-periodic points holds for every integer D>2, the result would be a natural and useful extension of Ingram’s D=2 theorem in the arithmetic dynamics of polynomial automorphisms. A parameter-controlled, effectively computable lower bound would also be of independent interest for questions of preperiodic points and Northcott-type finiteness for this family. The significance cannot be fully assessed from the abstract alone, since neither the strength of the constant nor the range of admissible B and base fields is stated.
major comments (3)
- The abstract asserts a lower bound for ĥ_φ of non-periodic points when D>2 but does not state the precise form of the bound, its dependence on D, B or the base field, or whether the constant is absolute or may tend to zero with D. Without an explicit statement, the central claim cannot be checked for uniformity or for residual degree-dependent error terms that may appear once the monomial Y^D replaces Y^2.
- The extension from Ingram’s D=2 estimates is presented as the main contribution, yet the abstract gives no indication that the local height comparisons, Green-function estimates, or cancellation of leading terms continue to produce a strictly positive constant independent of the point once deg(φ)=D>2. Control of those residuals is load-bearing for the claimed positivity; their absence from the abstract leaves the result unverifiable.
- No full text, proofs, or explicit constants are available for review. A referee report on correctness is therefore impossible; the manuscript must be supplied in complete form before any recommendation other than ‘uncertain’ can be issued.
minor comments (2)
- The abstract notation mixes B and b for the constant term of the map; a single consistent symbol should be fixed.
- The citation to Ingram’s D=2 work is given only as [Ingram1]; a full bibliographic reference would aid the reader.
Circularity Check
No circularity detected; abstract claims a non-tautological lower-bound extension of external prior work.
full rationale
Only the abstract is available. It states a lower bound on the canonical height ĥ_φ of non-periodic points for the Hénon maps φ(X,Y)=(Y,X+Y^D+B) when D>2, extending the D=2 case of Ingram. The canonical height is a standard, independently defined object in arithmetic dynamics; a uniform positive lower bound for non-periodic rational points is a contentful theorem, not equivalent by construction to the definition of ĥ_φ. The sole citation is to Ingram’s earlier work (external author, not a self-citation). No fitted parameters, no uniqueness theorems imported from the present author, no ansatz smuggled via self-citation, and no renaming of a known empirical pattern appear in the abstract. Skeptical concerns about whether degree-dependent error terms remain controlled for D>2 are questions of correctness, not circularity. With no equations or derivation steps supplied that reduce the claimed bound to its inputs, the honest finding is zero circularity.
Axiom & Free-Parameter Ledger
axioms (3)
- domain assumption Canonical height ĥ_φ associated to a Hénon map vanishes precisely on the preperiodic points and is non-negative.
- domain assumption The maps φ(X,Y)=(Y,X+Y^D+B) are polynomial automorphisms of the affine plane defined over a number field.
- ad hoc to paper Techniques that produce a positive lower bound when D=2 continue to work for every integer D>2.
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.
discussion (0)
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