Weighted linearization of vector fields via a formal Moser trick

TL;DR

This paper introduces a weighted linearization concept for vector fields, extending classical linearization theorems using a formal Moser trick applicable over any characteristic zero field.

Contribution

It formulates a weighted non-resonance condition and develops a formal Moser trick framework for weighted linearization of vector fields.

Findings

Weighted non-resonance condition implies weighted linearizability.

Develops a formal Moser trick applicable over any characteristic zero field.

Provides a general framework for weighted linearization on formal power series.

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.