Differential algebra and related topics : proceedings of the International Workshop, Newark Campus of Rutgers, the State University of New Jersey, 2-3 November 2000

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. Contents:the Ritt–kolchin Theory For Differential Polynomials (w Y Sit)differential Schemes (j J Kovacic)differential Algebra — A Scheme Theory Approach (h Gillet)model Theory And Differential Algebra (t Scanlon)inverse Differential Galois Theory (a R Magid)differential Galois Theory, Universal Rings And Universal Groups (m Van Der Put)cyclic Vectors (r C Churchill & J J Kovacic)differential Algebraic Techniques In Hamiltonian Mechanics (r C Churchill)moving Frames And Differential Algebra (e L Mansfield)baxter Algebras And Differential Algebras (l Guo) Readership: Graduate Students, Pure Mathematicians, Logicians, Algebraic Geometers, Applied Mathematicians And Physicists. Keywords:differential Algebra;mathematical Logic;algebraic Geometry;mathematical Physics 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