Involution. The formal theory of differential equations and its applications in computer algebra
type: Item
instance of: scholarly article
title: Involution. The formal theory of differential equations and its applications in computer algebra
zbMATH Open document ID: 1205.35003
DOI: 10.1007/978-3-642-01287-7
author: Werner M. Seiler
published in: Algorithms and Computation in Mathematics
publication date: 2009-05-26T00:00:00Z
review text: The book is a comprehensive monograph describing the formal theory of differential equations in theoretical and computational aspects. It is devoted to an exposition of classical and modern results on formal methods in the investigation of general differential equations, including ordinary and partial, overdetermined and underdetermined equations, non-normal systems, implicit differential-algebraic equations, etc. The author stresses the constructive nature of the methods and puts a special emphasis on algorithms that can be effectively implemented into symbolic software. The notion of involution in the book unites the algebraic, geometric and combinatorial aspects of the formal theory of differential equations. A significant part of the book is devoted to Pommaret bases, which are special kinds of Gröbner bases with specification for differential systems; in particular, they are useful when working with Koszul homology and Spencer cohomology. Cartan-Kähler theory, which is the general method to formulate a Cauchy problem, is presented in the book in its dual Vessiot version. These two features have mostly been presented in article form before, and so are described in this book in full detail for the first time. Let us present the contents of the book. The first chapter is an introduction that presents a range of sources for various types of differential equations and gives the motivation for a formal investigation of such equations. Different points of view of the mathematical and the physical community are underlined. Chapter 2 concerns the geometry of jet spaces and differential equations. The author chooses two approaches to this classical subject -- computational via coordinates and power series, and theoretical, where such important notions as affine structure in jet-fibers and prolongations are discussed. The third chapter elaborates upon the algebraic theory, which extends the classical Janet-Riquier theory. It introduces involutive division on a monoid (wit…
reviewed by: Boris Kruglikov
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13P20
Mathematics Subject Classification ID: 35-04
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 5557127
zbMATH Keywords: Gröbner basis
zbMATH Keywords: Pommaret basis
zbMATH Keywords: differential ideal
zbMATH Keywords: symbolic module
zbMATH Keywords: involutive system
zbMATH Keywords: formal integrability
zbMATH Keywords: Cartan-Kähler theory
zbMATH Keywords: Vessiot theory
zbMATH Keywords: Spencer cohomology
zbMATH Keywords: Koszul homology
zbMATH Keywords: overdetermined and underdetermined equations
zbMATH Keywords: non-normal systems
zbMATH Keywords: implicit differential-algebraic equations
describes a project that uses: EDS
describes a project that uses: MuPAD
describes a project that uses: SINGULAR
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-01287-7
OpenAlex ID: W4232970652
description: scientific article; zbMATH DE number 5557127