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On the invariance of super-linearization under polynomial automorphisms

Anmol Harshana, Mohamed-Ali Belabbas

TL;DR

Addresses whether the class $\mathcal{S}_n$ of super-linearizable polynomial vector fields is preserved under polynomial automorphisms. Develops a proof framework showing invariance under tame automorphisms (affine and elementary steps) and analyzes the weighted dependency graph (WDG) condition as a sufficient criterion for super-linearization, noting its noninvariance under some automorphisms. Shows that the WDG property is not generally preserved under elementary transformations, but lifting with a stabilizing observable can restore WDG compliance after a coordinate change. The results clarify how coordinate changes interact with lifted representations and guide the use of global linearization approaches in dynamical analysis, with implications for Koopman-based methods and polyflow representations.

Abstract

We prove that the super-linearizability of polynomial systems is preserved by all currently known classes of polynomial automorphisms of $\R^n$. We then establish connections between such automorphisms and a sufficient condition for super-linearizability.

On the invariance of super-linearization under polynomial automorphisms

TL;DR

Addresses whether the class of super-linearizable polynomial vector fields is preserved under polynomial automorphisms. Develops a proof framework showing invariance under tame automorphisms (affine and elementary steps) and analyzes the weighted dependency graph (WDG) condition as a sufficient criterion for super-linearization, noting its noninvariance under some automorphisms. Shows that the WDG property is not generally preserved under elementary transformations, but lifting with a stabilizing observable can restore WDG compliance after a coordinate change. The results clarify how coordinate changes interact with lifted representations and guide the use of global linearization approaches in dynamical analysis, with implications for Koopman-based methods and polyflow representations.

Abstract

We prove that the super-linearizability of polynomial systems is preserved by all currently known classes of polynomial automorphisms of . We then establish connections between such automorphisms and a sufficient condition for super-linearizability.

Paper Structure

This paper contains 10 sections, 10 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and Statement of the main results
  3. The WDG condition: a sufficient condition for super-linearization
  4. Proof of the main results
  5. Proof of Theorem \ref{['thm:tame']}
  6. Proof for Corollary \ref{['cor:stablytame']}: Case of Stably Tame Automorphisms
  7. WDG and Elementary transformations
  8. Conclusion and Future Work
  9. Proof of Lemma 2
  10. Non-superlinearizability of Example \ref{['eq:example_sinh']}

Key Result

Theorem 2.1

The class of polynomial super-linearizable systems is closed under transformation by tame automorphisms.

Figures (1)

  • Figure 1: A weighted dependency graph of a system that follows the sufficiency condition

Theorems & Definitions (25)

  • Definition 1: Lifted and Super-linearizable system
  • Definition 2: Affine Transformation
  • Definition 3: Elementary Transformation
  • Definition 4: Tame and Wild Automorphisms
  • Definition 5: Stably Tame Automorphisms
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3: WDG condition
  • Lemma 3.1
  • proof
  • ...and 15 more