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arxiv: 2604.04337 · v1 · submitted 2026-04-06 · 🧮 math.AG · math.AC

On the Tame Isotropy Group of Locally Finite Derivations of K[X,Y]

Luis Cid , Marcelo Veloso This is my paper

Pith reviewed 2026-05-10 20:18 UTC · model grok-4.3

classification 🧮 math.AG math.AC
keywords tame isotropy grouplocally finite derivationsK[X,Y]exponential automorphismpolynomial automorphismsVan den Essen classificationalgebraic geometry
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The pith

The tame isotropy group of any locally finite derivation on K[X,Y] equals the tame isotropy group of its exponential automorphism.

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

The paper proves that for locally finite derivations D of the polynomial ring K[X,Y] over an algebraically closed field of characteristic zero, the tame isotropy group Tame_D(K[X,Y]) always coincides with the tame isotropy group of the automorphism exp(D). This is shown by using Van den Essen's classification of such derivations into normal forms and computing the groups explicitly for each. A sympathetic reader cares because this equality provides a concrete way to understand stabilizers in the tame automorphism group, and it highlights a distinction from the full automorphism group where the corresponding equality can fail when the derivation has a nontrivial semisimple part.

Core claim

Using Van den Essen's classification of locally finite derivations up to conjugation, the tame isotropy group is determined explicitly for each normal form. It is then shown that this group always equals the tame isotropy group of the associated exponential automorphism exp(D). This contrasts with the behavior observed for the full automorphism group in the presence of a nontrivial semisimple part.

What carries the argument

The tame isotropy group Tame_D(K[X,Y]), the stabilizer of the derivation D inside the tame automorphism group of K[X,Y], compared directly to the corresponding stabilizer of exp(D).

Load-bearing premise

Van den Essen's classification provides a complete list of normal forms for locally finite derivations up to conjugation, allowing explicit computation of the isotropy groups for each form.

What would settle it

A specific locally finite derivation D on K[X,Y] for which the two tame isotropy groups differ, or a derivation outside the listed normal forms.

read the original abstract

Let K be an algebraically closed field of characteristic zero. We study the tame isotropy group Tame_D(K[X,Y]) of locally finite derivations of the polynomial ring K[X,Y], using Van den Essen's classification up to conjugation. For each normal form, we explicitly determine the corresponding tame isotropy group. We then compare Tame_D(K[X,Y]) with the tame isotropy group of the associated exponential automorphism exp(D), and prove that these groups always coincide. This stands in contrast to the behaviour of the full automorphism group, where such an equality may fail for derivations with a nontrivial semisimple part.

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

1 major / 2 minor

Summary. The paper studies the tame isotropy group Tame_D(K[X,Y]) for locally finite derivations D of the polynomial ring K[X,Y] over an algebraically closed field K of characteristic zero. Using Van den Essen's classification of such derivations up to conjugation, it explicitly determines the tame isotropy group for each normal form. It then proves that Tame_D(K[X,Y]) coincides with the tame isotropy group of the associated exponential automorphism exp(D) in every case, in contrast to the full automorphism group where equality can fail when the semisimple part is nontrivial.

Significance. If the explicit computations and equality hold, the result clarifies a structural distinction between the tame automorphism group and the full automorphism group Aut(K[X,Y]) with respect to isotropy under locally finite derivations. The case-by-case determination via normal forms supplies concrete group descriptions that may support further work on stabilizers, exponential maps, and the structure of automorphism groups in two-dimensional affine space.

major comments (1)
  1. The central proof reduces all cases to Van den Essen's normal forms and then computes the tame stabilizers explicitly before comparing them to those of exp(D). Because the manuscript invokes this external classification without reproducing the list of normal forms or the conjugation action in the text, it is difficult to verify that every locally finite derivation is covered and that no normal form was omitted in the comparison (see the paragraph following the statement of the main theorem).
minor comments (2)
  1. Notation for the tame isotropy group is introduced as Tame_D(K[X,Y]) but the precise definition (which elements of the tame automorphism group fix D) is not restated in the body; adding a short reminder would improve readability.
  2. The contrast with the full automorphism group is stated in the abstract and introduction but supported only by a reference to an earlier example; a one-sentence recap of that counter-example would make the distinction self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the recommendation for minor revision. The single major comment is addressed below.

read point-by-point responses

  1. Referee: The central proof reduces all cases to Van den Essen's normal forms and then computes the tame stabilizers explicitly before comparing them to those of exp(D). Because the manuscript invokes this external classification without reproducing the list of normal forms or the conjugation action in the text, it is difficult to verify that every locally finite derivation is covered and that no normal form was omitted in the comparison (see the paragraph following the statement of the main theorem).

    Authors: We agree that reproducing the normal forms would improve verifiability. In the revised manuscript we will insert, immediately after the statement of the main theorem, a self-contained paragraph that lists all normal forms appearing in Van den Essen's classification together with the explicit conjugation action that reduces an arbitrary locally finite derivation to one of these forms. This addition makes the exhaustive coverage of cases transparent while leaving the subsequent stabilizer computations unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper invokes Van den Essen's external classification of locally finite derivations to reduce to normal forms, then performs explicit case-by-case computation of the tame isotropy groups Tame_D(K[X,Y]) and compares them directly to the tame isotropy groups of the associated exp(D). This is a standard verification relying on an independent prior theorem, with no self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. The central equality is established by direct inspection in each normal form rather than by construction from the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the validity of Van den Essen's classification of locally finite derivations and the assumption that K is algebraically closed of characteristic zero.

axioms (1)
  • domain assumption Van den Essen's classification of locally finite derivations of K[X,Y] up to conjugation
    The paper uses this classification to determine the tame isotropy group for each normal form.

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discussion (0)

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

  1. [1]

    On the tame isotropy group of a derivation, 2025

    [AFV25] Angelo Bianchi Adriana Freitas and Marcelo Veloso. On the tame isotropy group of a derivation, 2025. [BM85] Hyman Bass and Gary Meisters. Polynomial flows in the plane.Advances in Mathematics, 55(2):173–208, 1985. [CV26] Luis Cid and Marcelo Veloso. On isotropy group of locally finite derivations onK[x, y], 2026. arXiv:2603.23709. [Mau03] Stefan M...

    work page arXiv 2025