arxiv: 2604.26950 · v2 · submitted 2026-04-29 · 🧮 math.DG · math.CA· math.DS· math.SG

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Weighted linearization of vector fields via a formal Moser trick

Arthur Lei Qiu

Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:20 UTC · model grok-4.3

classification 🧮 math.DG math.CAmath.DSmath.SG

keywords vector field linearizationweighted non-resonanceMoser trickformal power seriesPoincaré theoremSternberg theoremdifferential geometry

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The pith

A weighted non-resonance condition on eigenvalues implies formal weighted linearizability of vector fields.

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

By assigning weights to coordinates, one defines a weighted linear approximation to a vector field in which some directions count as linear and others as quadratic or higher. The paper formulates a weighted version of the classical non-resonance condition and proves that it guarantees a formal coordinate change reducing the vector field to this weighted linear part. The proof adapts Moser's trick to operate directly on formal power series expansions, and this version of the trick works over any field of characteristic zero.

Core claim

A suitably defined weighted non-resonance condition on the eigenvalues of the linear part implies that any vector field admitting a formal power series expansion is formally weighted-linearizable by a coordinate change obtained through a formal Moser trick.

What carries the argument

The formal Moser trick adapted to weighted power series, which solves the homological equation order by order according to the weighted grading to eliminate higher-order terms.

If this is right

The classical Poincaré and Sternberg linearization theorems appear as the special case in which all weights equal one.
Formal weighted linearization holds over any field of characteristic zero.
The result supplies a normal form for vector fields whose scaling is anisotropic with respect to the chosen weights.

Where Pith is reading between the lines

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

The same formal Moser trick may simplify resonance analysis in other normal-form problems that admit a natural weighted grading.
Convergence of the formal series to a smooth or analytic change of coordinates would require separate estimates not addressed here.
Weighted linearization could be tested numerically on low-dimensional examples with deliberately chosen weights to verify the formal procedure.

Load-bearing premise

The vector field admits a formal power series expansion and satisfies the weighted non-resonance condition so that the formal Moser trick encounters no further obstructions.

What would settle it

A concrete vector field whose eigenvalues meet the weighted non-resonance condition yet no formal power series change of coordinates reduces it to the corresponding weighted linear form.

read the original abstract

Many well-known theorems establish sufficient criteria for linearizability of a vector field in terms of the eigenvalues of its linear approximation. By attaching weights to coordinates so that some directions are considered "linear", others "quadratic", and so on, one can define the notion of a weighted linear approximation. It is thus natural to ask when a vector field is "weighted-linearizable". In this paper, we formulate a weighted version of the non-resonance condition appearing in the Poincar\'e and Sternberg linearization theorems and show that it implies weighted linearizability. Our approach first addresses weighted linearization on the level of formal power series. In doing so, we develop a general framework to make sense of a power series version of Moser's trick, a technique used to prove various normal form results in geometry. This formal Moser trick works over any field of characteristic zero and may be of independent interest.

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 gives a clean formal extension of linearization theorems to weighted vector fields via a general Moser trick in power series.

read the letter

The paper's core contribution is a weighted non-resonance condition that guarantees formal linearizability of a vector field via a Moser trick in power series. This extends the classical Poincaré and Sternberg theorems in a natural way for settings where coordinates have different weights. It does a good job setting up the formal Moser trick as a general tool. The trick solves the equation order by order using the weighted grading, and the proof works algebraically in characteristic zero without extra assumptions. This part looks reusable for other normal form problems in geometry. The central claim holds up in the formal setting. The non-resonance ensures invertibility of the linear operator on weighted homogeneous components, and the induction goes through cleanly with no circularity. The soft spot is the lack of any convergence results. Everything stays at the formal level, so it does not yet give actual diffeomorphisms in the analytic or C^infty category. That is a natural next step but not addressed here. This paper is aimed at specialists in dynamical systems and differential geometry who deal with weighted or homogeneous structures. It deserves a serious referee because the formal result is precise and the framework is new. I recommend putting it through peer review.

Referee Report

1 major / 3 minor

Summary. The manuscript claims that a suitably formulated weighted non-resonance condition on the eigenvalues of the linear part of a vector field implies the existence of a formal power-series diffeomorphism realizing weighted linearization. The argument proceeds by constructing a formal Moser trick that solves the homological equation order-by-order in the weighted grading; the non-resonance hypothesis guarantees invertibility of the relevant linear operator on spaces of weighted homogeneous vector fields over any field of characteristic zero.

Significance. If the result holds, it extends the classical Poincaré–Sternberg linearization theorems to weighted settings and supplies a general algebraic framework for formal Moser tricks that may be of independent interest for normal-form problems in geometry and dynamics. The purely formal, characteristic-zero construction without analytic or cohomological obstructions beyond the stated non-resonance is a clear strength.

major comments (1)

[§3] §3, around the statement of the weighted non-resonance condition and the homological operator: the proof that the operator is invertible on each weighted homogeneous component relies on the non-resonance hypothesis, but an explicit formula for the inverse (or at least a bound on the denominators) would make the inductive step fully transparent and rule out any characteristic-zero-specific cancellations.

minor comments (3)

[Abstract] Abstract: the sentence introducing the formal Moser trick would benefit from a parenthetical remark that the construction is purely algebraic and does not address convergence.
[§2] §2: the notation for weighted degrees and the associated filtration on the space of formal vector fields should be summarized in a single display for quick reference.
[Theorem 4.1] Theorem 4.1 (or the main result): the statement would be clearer if it explicitly records that the diffeomorphism is formal and tangent to the identity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestion. We have revised the manuscript to incorporate an explicit description of the inverse of the homological operator, which clarifies the inductive step.

read point-by-point responses

Referee: §3, around the statement of the weighted non-resonance condition and the homological operator: the proof that the operator is invertible on each weighted homogeneous component relies on the non-resonance hypothesis, but an explicit formula for the inverse (or at least a bound on the denominators) would make the inductive step fully transparent and rule out any characteristic-zero-specific cancellations.

Authors: We agree that an explicit formula improves transparency. In the revised §3 we now state that the homological operator acts diagonally on the monomial basis of weighted homogeneous vector fields: for a term x^α ∂_j the eigenvalue is ⟨λ, α⟩ − λ_j. The weighted non-resonance condition guarantees this quantity is nonzero in any field of characteristic zero; consequently the inverse is given by dividing the coefficient by this nonzero scalar. No further cancellations arise, and the formula makes the order-by-order solvability of the Moser equation immediate. We have added this description together with a short verification that the inverse preserves the weighted grading. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is algebraically self-contained

full rationale

The paper defines a weighted non-resonance condition on eigenvalues independently of any fitted quantities or prior results from the same authors, then constructs a formal Moser iteration that solves the homological equation order-by-order using only the algebraic invertibility of the linearized operator over characteristic-zero fields. This construction is presented as a general framework without invoking self-citations for uniqueness, without renaming known empirical patterns, and without smuggling ansatzes. The inductive step relies directly on the stated non-resonance hypothesis to guarantee solvability at each weighted degree, rendering the argument self-contained and free of reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard assumption that one works over a field of characteristic zero for formal power series manipulations, together with the definition of weighted degrees on coordinates.

axioms (1)

standard math Working over a field of characteristic zero
Required for the algebraic manipulations in the formal Moser trick to be well-defined.

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