Differential Algebra & Related Topics

✍ Scribed by P. Cassidy, Li Guo, William F. Keigher, Phyllis J. Cassidy, William Y. Sit

• Publisher: World Scientific Publishing Company
• Year: 2002
• Tongue: English
• Leaves: 321
• Edition: 1st
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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