On the structure of nilpotent normal form modules
type: Item
instance of: scholarly article
title: On the structure of nilpotent normal form modules
zbMATH Open document ID: 1012.34036
published in: Journal of Differential Equations
publication date: 2002-09-08T00:00:00Z
review text: The paper is concerned with the normal form theory of the differential equation \[ \dot{x}=Nx+a_2(x)+\cdots+a_j(x)+\cdots, \] where \(N\) is a nilpotent matrix, \(x\in\mathbb{R}^n\) and each \(a_j(x)\) is a homogeneous polynomial vector field of degree \(j\). An algorithm is presented that produces a Stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field given a Stanley decomposition for the ring of invariants. This reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm. Both the inner product normal form and the sl(2) normal form are covered within a unified theory, and simplified (nonequivariant) versions of each normal form are presented.
reviewed by: Ladislav Adamec
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 1797541
zbMATH Keywords: normal forms
zbMATH Keywords: Stanley decomposition
zbMATH Keywords: equivariants
author: James A. Murdock
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4058
OpenAlex ID: W1998751687
cites work: EQUIVALENCE AND NORMAL FORMS OF GERMS OF SMOOTH MAPPINGS
cites work: Q3197042
cites work: Normal form for the (2;n)-nilpotent vector field, using invariant theory
cites work: Q4317713
cites work: A simple global characterization for normal forms of singular vector fields
cites work: Q4002278
cites work: Asymptotic unfoldings of dynamical systems by normalizing beyond the normal form
cites work: Q4509277
cites work: Q4288567
cites work: Algorithms in invariant theory
cites work: Computing combinatorial decompositions of rings
cites work: Q3833289
DOI: 10.1006/JDEQ.2001.4058
description: scientific article; zbMATH DE number 1797541