$\mathbb T$-homogeneous locally nilpotent derivations of trinomial algebras

Kirill Rassolov; Timofey Krasikov

arxiv: 2605.17670 · v2 · pith:M7SCCEOVnew · submitted 2026-05-17 · 🧮 math.AG · math.AC

mathbb T-homogeneous locally nilpotent derivations of trinomial algebras

Timofey Krasikov , Kirill Rassolov This is my paper

Pith reviewed 2026-05-20 12:26 UTC · model grok-4.3

classification 🧮 math.AG math.AC

keywords trinomial algebraslocally nilpotent derivationstorus actioncomplexity oneCox ringsaffine varietiesalgebraic derivations

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The pith

The locally nilpotent derivations of trinomial algebras that are homogeneous under the natural torus action are fully described.

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

The paper establishes a description of all locally nilpotent derivations on trinomial algebras which remain homogeneous with respect to the torus action. Trinomial algebras are defined by compatible systems of three-term polynomial relations and function as the Cox rings of varieties that have torus actions of complexity one. This description matters because these derivations generate flows or automorphisms that preserve the torus symmetry on the variety. A reader would care as it provides concrete algebraic tools to analyze the geometry of these special varieties.

Core claim

We describe locally nilpotent derivations of a trinomial algebra that are homogeneous under a natural torus action of complexity one. The description is achieved by analyzing how the derivation acts on the generators of the algebra while respecting the three-term relations and the grading.

What carries the argument

T-homogeneous locally nilpotent derivation, which is a derivation that is locally nilpotent, commutes with the torus action in the homogeneous sense, and thus can be classified using the structure of the trinomial relations.

If this is right

One can explicitly construct all such derivations from the data of the relations.
These derivations correspond to G_a-actions on the associated affine variety that commute with the torus action.
The classification helps determine the structure of the automorphism group of the variety.
Applications include studying the orbits and invariants under these actions.

Where Pith is reading between the lines

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

The method may extend to studying derivations on algebras with relations of more terms.
This classification could be used to verify properties of specific varieties like del Pezzo surfaces or other examples with torus actions.
Connections to the broader theory of complexity-one varieties and their birational geometry might be explored.
It opens the door to computational checks in low degrees to see if all possible derivations are captured.

Load-bearing premise

The relations are compatible three-term polynomials that make the algebra finitely generated as the Cox ring of a complexity-one torus variety.

What would settle it

Construct a trinomial algebra with a specific T-homogeneous locally nilpotent derivation and check if it matches one of the forms given in the description; mismatch would falsify the completeness of the description.

read the original abstract

A trinomial algebra is a commutative finitely generated algebra given by a system of compatible relations each of which is a polynomial with three terms. Such algebras arise as the Cox rings of varieties admitting a complexity one torus action. We describe locally nilpotent derivations of a trinomial algebra that are homogeneous under a natural torus action of complexity one.

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.

Desk Editor's Note private letter to a colleague

The paper classifies T-homogeneous locally nilpotent derivations on trinomial algebras from compatible three-term relations, but the compatibility step stays largely assumptive.

read the letter

The core contribution is a description of the T-homogeneous locally nilpotent derivations on trinomial algebras. These algebras are defined by systems of three-term relations and appear as Cox rings for varieties with complexity-one torus actions. The authors focus on derivations that respect the natural torus grading and give an explicit form for them under that setup. That targeted classification is the main new piece; it narrows earlier results on graded derivations to this concrete class of algebras. If the proofs proceed by direct computation on the generators and relations, that approach fits the low-complexity setting and can be checked case by case. The paper also ties the derivations back to the geometry of the torus action, which helps readers see potential uses for automorphism questions. The writing stays technical but readable for people already working in Cox rings or torus actions. The main soft spot is the standing assumption that the three-term relations are compatible and produce a finitely generated algebra. The abstract states this as a hypothesis, yet the provided text does not include a general test or algorithm for verifying compatibility, such as degree bounds or Gröbner-basis checks that would survive the torus grading. Without that, the classification applies only after the user has already confirmed finite generation by other means. That is a real limitation for broader use, though it is common in classification papers that start from a fixed class of algebras. Minor points include whether all possible homogeneous derivations are captured or if some degenerate cases are omitted; those would need checking in the proofs. This work is aimed at specialists in algebraic geometry who study automorphism groups or explicit descriptions of Cox rings for torus actions of low complexity. A reader already familiar with the literature on graded derivations and complexity-one actions will find the explicit list useful for computations. It is narrow enough that it will not reshape the wider theory, but the concrete result is worth having on record. The paper shows clear engagement with the relevant setup and literature, so it deserves a serious referee even if revisions are needed on the compatibility criterion and examples. I would send it to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript defines trinomial algebras as commutative finitely generated k-algebras presented by a system of compatible three-term relations; these arise as Cox rings of varieties admitting a complexity-one torus action. It then classifies or describes the locally nilpotent derivations that are homogeneous with respect to the natural torus action of complexity one.

Significance. If the description is complete and correct, the work supplies an explicit handle on homogeneous LNDs on a class of graded algebras that appear as Cox rings, which is useful for studying automorphism groups, Demazure roots, and birational geometry of varieties with torus actions of complexity one. The homogeneity assumption aligns with standard techniques in the field and could lead to concrete applications once verified on examples.

major comments (1)

[§2] §2 (Definition of trinomial algebra and compatibility): The central claim presupposes that the system of three-term relations is compatible, which is required for the algebra to be finitely generated and to arise as a Cox ring. No general criterion, algorithm, or set of sufficient conditions (e.g., via Gröbner bases, degree bounds, or explicit checks on the relations) is supplied to verify compatibility for arbitrary choices. This assumption is load-bearing; if compatibility fails for a given set of relations, the algebra is not finitely generated and the subsequent classification of homogeneous LNDs does not apply.

minor comments (2)

[§1] Notation for the torus action and the grading should be introduced with a short table or explicit list of weights to improve readability.
[Introduction] Several statements in the introduction refer to 'the natural torus action' without a forward reference to the precise definition in §3; add a cross-reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of the compatibility assumption in the definition of trinomial algebras. We respond to the major comment below.

read point-by-point responses

Referee: [§2] §2 (Definition of trinomial algebra and compatibility): The central claim presupposes that the system of three-term relations is compatible, which is required for the algebra to be finitely generated and to arise as a Cox ring. No general criterion, algorithm, or set of sufficient conditions (e.g., via Gröbner bases, degree bounds, or explicit checks on the relations) is supplied to verify compatibility for arbitrary choices. This assumption is load-bearing; if compatibility fails for a given set of relations, the algebra is not finitely generated and the subsequent classification of homogeneous LNDs does not apply.

Authors: We agree that compatibility of the relations is essential for finite generation and for the algebra to arise as a Cox ring of a variety with a complexity-one torus action. The manuscript defines a trinomial algebra as a commutative finitely generated k-algebra presented by a system of compatible three-term relations; our description of T-homogeneous locally nilpotent derivations is stated for precisely these algebras. We do not supply a general criterion or algorithm for verifying compatibility because the question of whether an arbitrary set of three-term relations generates a finitely generated algebra is a broad computational problem that depends on the specific degrees and coefficients and is not the focus of this work. In the geometric setting from which these algebras arise, compatibility follows from the construction of the Cox ring. For concrete examples, compatibility is typically verified by direct computation or by reference to known presentations in the literature on complexity-one quotients. We will add a clarifying paragraph in §2 emphasizing that all statements assume the given relations are compatible (as required by the definition) and that the results do not apply when this fails. This is a partial revision that strengthens the exposition without altering the scope or adding a general verification procedure. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation proceeds from external algebraic hypothesis

full rationale

The paper posits compatibility of three-term relations as the hypothesis ensuring finite generation of the trinomial algebra as a Cox ring for a complexity-one torus action. From this standard setup in algebraic geometry, it then classifies the T-homogeneous locally nilpotent derivations. No equations, self-citations, fitted parameters renamed as predictions, or ansatzes are visible in the abstract or described structure that would reduce the central claim back to its inputs by construction. The assumption is load-bearing but external and falsifiable independently of the derivation, so the chain is self-contained against benchmarks for Cox rings and LNDs on graded algebras.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of trinomial algebras via compatible three-term relations and their identification with Cox rings of complexity-one torus actions; no free parameters or invented entities are visible in the abstract.

axioms (2)

domain assumption Trinomial algebras are commutative finitely generated algebras given by a system of compatible three-term polynomial relations.
Stated in the first sentence of the abstract as the object of study.
domain assumption Such algebras arise as the Cox rings of varieties admitting a complexity one torus action.
Stated in the second sentence of the abstract.

pith-pipeline@v0.9.0 · 5572 in / 1077 out tokens · 34052 ms · 2026-05-20T12:26:27.299799+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

A trinomial algebra is a commutative finitely generated algebra given by a system of compatible relations each of which is a polynomial with three terms. Such algebras arise as the Cox rings of varieties admitting a complexity one torus action.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We describe locally nilpotent derivations of a trinomial algebra that are homogeneous under a natural torus action of complexity one.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

[1]

Ann.334(2006), 3, 557–607

Altmann K., Hausen J.,Polyhedral divisors and algebraic torus actions, Math. Ann.334(2006), 3, 557–607

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V.,On factoriality of Cox rings, Math

Arzhantsev I. V.,On factoriality of Cox rings, Math. Notes85(2009), 5, 623–629. T-HOMOGENEOUS LOCALLY NILPOTENT DERIV ATIONS OF TRINOMIAL ALGEBRAS 21

work page 2009
[3]

Arzhantsev I.,On rigidity of factorial trinomial hypersurfaces, Internat. J. Algebra Comput.26(2016), 5, 1061–1070

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Arzhantsev I., Derenthal U., Hausen J., Laface A.,Cox Rings, Cambridge Studies in Advanced Mathematics 144, Cambridge Univ. Press, Cambridge, 2014

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work page 2012
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Cox D.,The homogeneous coordinate ring of a toric variety, J. Algebraic Geom.4(1995), 1, 17–50

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Cox D., Little J., Schenck H.Toric Varieties. Grad. Stud. Math.124, AMS, Providence, RI, 2011

work page 2011
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Demazure M.,Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér.3(1970), 4, 507–588

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A,Rigid Trinomial Varieties, arXiv:2307.06672v1, 2023

Evdokimova P., Gaifullin S., Shafarevich. A,Rigid Trinomial Varieties, arXiv:2307.06672v1, 2023

work page arXiv 2023
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Sci.136, Springer- Verlag, Berlin, Heidelberg, 2017

Freudenburg G.,Algebraic theory of locally nilpotent derivations, Encyclopaedia Math. Sci.136, Springer- Verlag, Berlin, Heidelberg, 2017

work page 2017
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Algebra573(2021), 364–392

Gaifullin S.,Automorphisms of Danielewski varieties, J. Algebra573(2021), 364–392

work page 2021
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Gaifullin S.,Rigidity of trinomial hypersurfaces and factorial trinomial varieties, arXiv:1902.06136 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1902
[13]

Algebra Appl.18(2019), 10, 1–19

Gaifullin S., Zaitseva Y.,On homogeneous locally nilpotent derivations of trinomial algebras, J. Algebra Appl.18(2019), 10, 1–19

work page 2019
[14]

In: Torsors, Étale Homotopy and Applications to Rational Points, London Math

Hausen J., Herppich E.,Factorially graded rings of complexity one. In: Torsors, Étale Homotopy and Applications to Rational Points, London Math. Soc. Lecture Note Ser.405, Cambridge Univ. Press, Cambridge, 2013, 414–428

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Hausen J., Wrobel M.,Non-complete rational T-varieties of complexity one, Math. Nachr.290(2017), 5-6, 815–826

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Groups.15 (2010), 2, 389–425

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and Vinberg E.,Invariant Theory.In: Parshin, A.N., Shafarevich, I.R

Popov V. and Vinberg E.,Invariant Theory.In: Parshin, A.N., Shafarevich, I.R. (eds) Algebraic Geometry IV. Encyclopaedia of Mathematical Sciences, vol 55. Springer, Berlin, Heidelberg (1994)

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A,Homogeneous locally nilpotent derivations on trinomial varieties, Math

Rassolov K. A,Homogeneous locally nilpotent derivations on trinomial varieties, Math. Notes118(2025), 4, 820–835

work page 2025
[20]

Notes105 (2019), 6, 818–830

Zaitseva Y.,Homogeneous locally nilpotent derivations of nonfactorial trinomial algebras, Math. Notes105 (2019), 6, 818–830. Timofey Krasikov E-mail:timkrasikov@gmail.com Kirill Rassolov E-mail:kirill.rassolov@math.msu.ru Lomonosov Moscow State University, F aculty of Mechanics and Mathematics, Department of Higher Algebra, Leninskie Gory 1, Moscow, 11999...

work page 2019

[1] [1]

Ann.334(2006), 3, 557–607

Altmann K., Hausen J.,Polyhedral divisors and algebraic torus actions, Math. Ann.334(2006), 3, 557–607

work page 2006

[2] [2]

V.,On factoriality of Cox rings, Math

Arzhantsev I. V.,On factoriality of Cox rings, Math. Notes85(2009), 5, 623–629. T-HOMOGENEOUS LOCALLY NILPOTENT DERIV ATIONS OF TRINOMIAL ALGEBRAS 21

work page 2009

[3] [3]

Arzhantsev I.,On rigidity of factorial trinomial hypersurfaces, Internat. J. Algebra Comput.26(2016), 5, 1061–1070

work page 2016

[4] [4]

Press, Cambridge, 2014

Arzhantsev I., Derenthal U., Hausen J., Laface A.,Cox Rings, Cambridge Studies in Advanced Mathematics 144, Cambridge Univ. Press, Cambridge, 2014

work page 2014

[5] [5]

J.61 (2012), 4, 731–762

Arzhantsev I., Liendo A.,Polyhedral divisors andSL2-actions on affineT-varieties, Michigan Math. J.61 (2012), 4, 731–762

work page 2012

[6] [6]

Algebraic Geom.4(1995), 1, 17–50

Cox D.,The homogeneous coordinate ring of a toric variety, J. Algebraic Geom.4(1995), 1, 17–50

work page 1995

[7] [7]

Cox D., Little J., Schenck H.Toric Varieties. Grad. Stud. Math.124, AMS, Providence, RI, 2011

work page 2011

[8] [8]

Demazure M.,Sous-groupes algébriques de rang maximum du groupe de Cremona, Ann. Sci. Éc. Norm. Supér.3(1970), 4, 507–588

work page 1970

[9] [9]

A,Rigid Trinomial Varieties, arXiv:2307.06672v1, 2023

Evdokimova P., Gaifullin S., Shafarevich. A,Rigid Trinomial Varieties, arXiv:2307.06672v1, 2023

work page arXiv 2023

[10] [10]

Sci.136, Springer- Verlag, Berlin, Heidelberg, 2017

Freudenburg G.,Algebraic theory of locally nilpotent derivations, Encyclopaedia Math. Sci.136, Springer- Verlag, Berlin, Heidelberg, 2017

work page 2017

[11] [11]

Algebra573(2021), 364–392

Gaifullin S.,Automorphisms of Danielewski varieties, J. Algebra573(2021), 364–392

work page 2021

[12] [12]

Gaifullin S.,Rigidity of trinomial hypersurfaces and factorial trinomial varieties, arXiv:1902.06136 (2019)

work page internal anchor Pith review Pith/arXiv arXiv 1902

[13] [13]

Algebra Appl.18(2019), 10, 1–19

Gaifullin S., Zaitseva Y.,On homogeneous locally nilpotent derivations of trinomial algebras, J. Algebra Appl.18(2019), 10, 1–19

work page 2019

[14] [14]

In: Torsors, Étale Homotopy and Applications to Rational Points, London Math

Hausen J., Herppich E.,Factorially graded rings of complexity one. In: Torsors, Étale Homotopy and Applications to Rational Points, London Math. Soc. Lecture Note Ser.405, Cambridge Univ. Press, Cambridge, 2013, 414–428

work page 2013

[15] [15]

Nachr.290(2017), 5-6, 815–826

Hausen J., Wrobel M.,Non-complete rational T-varieties of complexity one, Math. Nachr.290(2017), 5-6, 815–826

work page 2017

[16] [16]

Algebra Geom.55(2014), 2, 621–634

Kotenkova P.,On restriction of roots on affineT-varieties, Beitr. Algebra Geom.55(2014), 2, 621–634

work page 2014

[17] [17]

Groups.15 (2010), 2, 389–425

Liendo A.,AffineT-varieties of complexity one and locally nilpotent derivations, Transform. Groups.15 (2010), 2, 389–425

work page 2010

[18] [18]

and Vinberg E.,Invariant Theory.In: Parshin, A.N., Shafarevich, I.R

Popov V. and Vinberg E.,Invariant Theory.In: Parshin, A.N., Shafarevich, I.R. (eds) Algebraic Geometry IV. Encyclopaedia of Mathematical Sciences, vol 55. Springer, Berlin, Heidelberg (1994)

work page 1994

[19] [19]

A,Homogeneous locally nilpotent derivations on trinomial varieties, Math

Rassolov K. A,Homogeneous locally nilpotent derivations on trinomial varieties, Math. Notes118(2025), 4, 820–835

work page 2025

[20] [20]

Notes105 (2019), 6, 818–830

Zaitseva Y.,Homogeneous locally nilpotent derivations of nonfactorial trinomial algebras, Math. Notes105 (2019), 6, 818–830. Timofey Krasikov E-mail:timkrasikov@gmail.com Kirill Rassolov E-mail:kirill.rassolov@math.msu.ru Lomonosov Moscow State University, F aculty of Mechanics and Mathematics, Department of Higher Algebra, Leninskie Gory 1, Moscow, 11999...

work page 2019