Differential algebra explores properties of solutions to systems of (ordinary or partial, linear or nonlinear) differential equations from an algebraic point of view. It includes as special cases algebraic systems as well as differential systems with algebraic constraints. This algebraic theory of Joseph F Ritt and Ellis R Kolchin is further enriched by its interactions with algebraic geometry, Diophantine geometry, differential geometry, model theory, control theory, automatic theorem proving, combinatorics, and difference equations. Differential algebra now plays an important role in computational methods such as symbolic integration, and symmetry analysis of differential equations. This volume includes tutorial and survey papers presented at workshop.
Readership: Graduate students, pure mathematicians, logicians, algebraic geometers, applied mathematicians and physicists.
Author(s): P. Cassidy, Li Guo, William F. Keigher, Phyllis J. Cassidy, William Y. Sit
Edition: 1st
Publisher: World Scientific Publishing Company
Year: 2002
Language: English
Pages: C+xiii+306+B
Cover
S Title
Differential Algebra and Related Topics
Copyright © 2002 by World Scientific
ISBN 981-02-4703-6
Foreword
Acknowledgements
Workshop Participants
Workshop Program
Contents
THE RITT-KOLCHIN THEORY FOR DIFFERENTIAL POLYNOMIALS
Preface
1 Basic Definitions
2 Triangular Sets and Pseudo-Division
3 Invertibility of Initials
4 Ranking and Reduction Concepts
5 Characteristic Sets
6 Reduction Algorithms
7 Rosenfeld Properties of an Autoreduced Set
8 Coherence and Rosenfeld's Lemma
9 Ritt-Raudenbush Basis Theorem
10 Decomposition Problems
11 Component Theorems
12 The Low Power Theorem
Appendix: Solutions and hints to selected exercises
Acknowledgements
References
DIFFERENTIAL SCHEMES
1 Introduction
2 Differential rings
3 Differential spectrum
4 Structure sheaf
5 Morphisms
6 Delta\ -Schemes
7 \Delta -Zeros
8 Differential spectrum of R
9 AAD modules
10 Global sections of AAD rings
11 AAD schemes
12 AAD reduction
13 Based schemes
14 Products
References
DIFFERENTIAL ALGEBRA: A SCHEME THEORY APPROACH
Introduction
1 Differential Rings
1.1 Some Commutative Algebra
1.2 Prolongation
2 Kolchin's Irreducibility Theorem
3 Descent for Projective Varieties
3.1 Proof of the theorem
3.2 Remark
4 Complements and Questions
4.1 Hasse-Schmidt Differentiation
4.2 Derivations and Valuation Rings
References
MODEL THEORY AND DIFFERENTIAL ALGEBRA
1 Introduction
2 Notation and conventions in differential algebra
3 What is model theory?
4 Differentially closed fields
4.1 Universal domains and quantifier elimination
4.2 Totally transcendental theories, Zariski geometries, and ranks
4.3 Generalized differential Galois theory
4.4 Classification of trivial differential equations
4.5 Differential fields of positive characteristic
5 0-minimal theories
6 Valued differential fields
7 Model theory of difference fields
Acknowledgments
References
INVERSE DIFFERENTIAL GALOIS THEORY
1 Introduction
1.1 Picard- Vessiot extensions
1.2 Statement of the inverse problem
2 The derivation approach to the inverse problem
3 The inverse problem for a 2 x 2 upper triangular matrix group
4 Solvable groups
4.1 Tori
4.2 Unipotents
4.3 General solvable case
Acknowledgments
References
DIFFERENTIAL GALOIS THEORY, UNIVERSAL RINGS AND UNIVERSAL GROUPS
1 The basic concepts
2 Universal Picard-Vessiot rings
2.1 The formalism of affine group schemes
2.2 Classes of differential modules
3 Regular singular equations
4 Formal differential equations
5 Multisummation and Stokes maps
6 Meromorphic differential equations
6.1 A construction with free Lie algebras
References
CYCLIC VECTORS
1 Introduction
2 Linear Differential Equations
3 The Algorithm
4 Remarks on the Algorithm
5 Remarks on the Hypotheses
6 Counterexamples
7 An Alternate Approach
Acknowledgment
Appendix - A MAPLE implementation
References
DIFFERENTIAL ALGEBRAIC TECHNIQUES IN HAMILTONIAN DYNAMICS
1 Integrals of Ordinary Differential Equations
2 Linearized Equations
3 Hamiltonian Systems - The Classical Formulation
4 Normal Variational Equations
5 Differential Galois Theory and Non-Integrability
6 Preliminaries to the Applications
7 Applications
References
MOVING FRAMES AND DIFFERENTIAL ALGEBRA
Introduction
1 Moving frames, a tutorial
1.1 Group Actions and differential invariants
1.2 Constructing moving frames
1.3 Construction of differential invariants
1.4 Applications
2 Comparison of {u_K||K|> 0} with {I_k I IK|> 0}
3 Calculations with invariants
Acknowledgments
References
BAXTER ALGEBRAS AND DIFFERENTIAL ALGEBRAS
0 Introduction
0.1 Relation with differential algebra
0.2 Some history
0.3 Outline
1 Definitions, examples and basic properties
1.1 Definitions and examples
1.2 Integrations and summations
2 Free Baxter algebras
2.1 Free Baxter algebras of Cartier
2.2 Mixable shuffle Baxter algebras
2.3 Standard Baxter algebras
3 Further applications of free Baxter algebras
3.1 Overview
3.2 Hopf algebra
3.3 The uinbral calculus
References
Back Cover
i-xiii
FRONT MATTER
1-70
THE RITT–KOLCHIN THEORY FOR DIFFERENTIAL POLYNOMIALS
WILLIAM Y. SIT
Abstract
71-94
DIFFERENTIAL SCHEMES
JERALD J. KOVACIC
95-124
DIFFERENTIAL ALGEBRA A SCHEME THEORY APPROACH
HENRI GILLET
125-150
MODEL THEORY AND DIFFERENTIAL ALGEBRA
THOMAS SCANLON
151-170
INVERSE DIFFERENTIAL GALOIS THEORY
ANDY R. MAGID
171-190
DIFFERENTIAL GALOIS THEORY, UNIVERSAL RINGS AND UNIVERSAL GROUPS
MARIUS VAN DER PUT
191-218
CYCLIC VECTORS
R. C. CHURCHILL, JERALD J. KOVACIC
219-256
DIFFERENTIAL ALGEBRAIC TECHNIQUES IN HAMILTONIAN DYNAMICS
RICHARD C. CHURCHILL
257
257-280
MOVING FRAMES AND DIFFERENTIAL ALGEBRA
ELIZABETH L. MANSFIELD
281-305
BAXTER ALGEBRAS AND DIFFERENTIAL ALGEBRAS
LI GUO
