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arxiv: 1906.11188 · v1 · pith:NSSHCYDKnew · submitted 2019-06-26 · 🧮 math.NT · math.AG· math.DS

Higher arithmetic degrees of dominant rational self-maps

Nguyen-Bac Dang , Dragos Ghioca , Fei Hu , John Lesieutre , Matthew Satriano This is my paper

Pith reviewed 2026-05-25 15:07 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.DS
keywords arithmetic degreesdynamical degreesKawaguchi-Silverman conjecturedominant rational mapsprojective varietiesheight growthorbits of cycles
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The pith

Arithmetic degrees are defined for orbits of higher-dimensional cycles and conjectured to equal dynamical degrees.

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

The paper extends the Kawaguchi-Silverman conjecture, which links height growth along point orbits to the first dynamical degree, to the orbits of higher-dimensional subvarieties. It defines a collection of arithmetic degrees attached to a dominant rational self-map that measure height growth along the forward orbit of any cycle and proves this value does not depend on the particular cycle. The authors then build a parallel body of results for these arithmetic degrees that mirrors known facts about dynamical degrees. Several conjectures are stated that relate the new arithmetic degrees to the dynamical degrees of the map.

Core claim

We define a set of arithmetic degrees of f, independent of the choice of cycle, and develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

What carries the argument

The arithmetic degree of a dominant rational self-map with respect to a cycle, given by the limit of normalized height growth along the forward orbit of the cycle.

If this is right

  • The arithmetic degree attached to any cycle is independent of the cycle chosen.
  • Properties of arithmetic degrees hold in direct analogy with those already known for dynamical degrees.
  • The extended Kawaguchi-Silverman conjecture asserts that the arithmetic degree equals the dynamical degree whenever the orbit is dense in the variety.
  • The higher arithmetic degrees are governed by a set of conjectural relations to the dynamical degrees of the map.

Where Pith is reading between the lines

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

  • Verification on low-dimensional examples such as maps on projective space would provide immediate tests of the conjectures.
  • The definitions open the possibility of using height computations on cycles to obtain information about dynamical degrees.
  • The parallel development suggests that other results known for dynamical degrees may admit direct arithmetic counterparts.

Load-bearing premise

The limit that defines the arithmetic degree along the orbit of a higher-dimensional cycle exists and can be compared directly to the dynamical degree.

What would settle it

An explicit dominant rational map on a smooth projective variety over the algebraic closure of the rationals, together with a cycle whose forward orbit is dense, for which the height growth rate differs from the first dynamical degree.

read the original abstract

Suppose that $f \colon X \dashrightarrow X$ is a dominant rational self-map of a smooth projective variety defined over ${\overline{\mathbf Q}}$. Kawaguchi and Silverman conjectured that if $P \in X({\overline{\mathbf Q}})$ is a point with well-defined forward orbit, then the growth rate of the height along the orbit exists, and coincides with the first dynamical degree $\lambda_1(f)$ of $f$ if the orbit of $P$ is Zariski dense in $X$. In this note, we extend the Kawaguchi-Silverman conjecture to the setting of orbits of higher-dimensional subvarieties of $X$. We begin by defining a set of arithmetic degrees of $f$, independent of the choice of cycle, and we then develop the theory of arithmetic degrees in parallel to existing results for dynamical degrees. We formulate several conjectures governing these higher arithmetic degrees, relating them to dynamical degrees.

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 / 2 minor

Summary. The manuscript extends the Kawaguchi-Silverman conjecture from points to orbits of higher-dimensional subvarieties by introducing higher arithmetic degrees for a dominant rational self-map f : X ⇢ X. These degrees are defined so as to be independent of the choice of cycle, the theory is developed in parallel with existing results on dynamical degrees, and several conjectures are formulated that relate the higher arithmetic degrees to the dynamical degrees of f.

Significance. If the conjectures hold, the framework supplies a coherent higher-dimensional extension of arithmetic dynamics, permitting the comparison of height growth rates along subvariety orbits with dynamical degrees in a manner directly analogous to the point case. The explicit labeling of the conjectures and the construction of cycle-independent quantities constitute the main strengths of the note.

minor comments (2)
  1. The abstract states that the arithmetic degrees are independent of the choice of cycle; the corresponding definition section should include a short explicit verification that the limit (when it exists) is indeed invariant under rational equivalence or linear equivalence of cycles.
  2. Number the conjectures (e.g., Conjecture 3.1, Conjecture 3.2) so that later references to them are unambiguous.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; purely definitional and conjectural

full rationale

The manuscript defines arithmetic degrees for cycles (claimed independent of representative) and formulates explicit conjectures paralleling known dynamical-degree results. No theorems are proved whose validity would require the arithmetic-degree limit to exist a priori; existence is instead packaged inside the conjectures. No equations reduce a claimed prediction to a fitted input by construction, no self-citation chain bears the central claim, and no ansatz is smuggled via prior work by the same authors. The derivation chain is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on standard background from algebraic geometry and arithmetic dynamics; no free parameters appear because the contribution is definitional and conjectural rather than computational or data-driven.

axioms (1)
  • standard math Existence and basic properties of heights and dynamical degrees on projective varieties over algebraic closures of Q
    The extension and parallel theory presuppose these established objects from prior literature.
invented entities (1)
  • Higher arithmetic degrees no independent evidence
    purpose: Quantities measuring height growth along subvariety orbits, independent of cycle representative
    Newly introduced objects whose existence and relations are conjectured rather than derived from external evidence.

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Reference graph

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