Approximate Commutative Algebra (eBook)

Foreword 6
Preface 7
Contents 11
From Oil Fields to Hilbert Schemes 15
Contents 16
Introduction 16
1.1 A Problem Arising in Industrial Mathematics 19
1.2 Border Bases 24
1.3 The Eigenvalue Method for Solving Polynomial Systems 32
1.4 Approximate Vanishing Ideals 36
1.5 Stable Order Ideals 45
1.6 Border Basis and Gröbner Basis Schemes 54
References 67
Numerical Decomposition of the Rank- Deficiency Set of a Matrix of Multivariate Polynomials 69
Introduction 70
2.1 Background Material 73
2.2 Random Coordinate Patches on Grassmannians 77
2.3 Finding Rank-Dropping Sets 79
2.4 Generalizations 81
2.5 Applications 83
2.6 Implementation Details and Computational Results 85
2.7 The Singular Set of the Reduction of an Algebraic Set 86
References 89
Towards Geometric Completion of Differential Systems by Points 93
3.1 Introduction 94
3.2 Zero Sets of PDE 96
3.3 Witness Sets of PDE 97
3.4 Geometric Lifting and Singular Components 100
3.5 Determination of Singular Components of an ODE using Numerical Jet Geometry 102
3.6 Determination of Singular Components of a PDE System 104
3.7 Discussion 108
Acknowledgement 109
References 109
Geometric Involutive Bases and Applications to Approximate Commutative Algebra 113
Introduction 113
4.1 Jet Spaces and Geometric Involutive Bases 116
4.2 Geometric Projected Involutive Bases and Nearby Systems 121
4.3 The Hilbert Function 126
4.4 Applications 128
4.5 Appendix 133
References 137
Regularization and Matrix Computation in Numerical Polynomial Algebra 139
Introduction 139
5.1 Notation and preliminaries 141
5.2 Formulation of the approximate solution 148
5.3 Matrix computation arising in polynomial algebra 156
5.4 A subspace strategy for efficient matrix computations 164
5.5 Software development 169
References 172
Ideal Interpolation: Translations to and from Algebraic Geometry 177
6.1 Introduction 177
6.2 Hermite Projectors and Their Relatives 184
6.3 Nested Ideal Interpolation 194
6.4 Error Formula 199
6.5 Loss of Haar 201
Acknowledgment 203
Appendix: AT-AG dictionary 204
References 204
An Introduction to Regression and Errors in Variables from an Algebraic Viewpoint 207
7.1 Regression and the X -matrix 207
7.2 Orthogonal polynomials and the residual space 210
7.3 The fitted function and its variance 212
7.4 “Errors in variables” analysis of polynomial models 213
7.5 Comments 215
7.6 Acknowledgements 216
References 216
ApCoA = Embedding Commutative Algebra into Analysis 219
8.1 Introduction 219
8.2 Approximate Commutative Algebra 220
8.3 Empirical Data 221
8.4 Valid Results Validity Checking of Results
8.5 Data . Result Mappings 223
8.6 Analytic View of Data.Result Mappings 224
8.7 Condition 225
8.8 Overdetermination 227
8.9 Syzygies 228
8.10 Singularities 229
8.11 Conclusions 231
References 231
Exact Certification in Global Polynomial Optimization Via Rationalizing Sums- Of- Squares 233
Narrative 233
References 239

Chapter 3 Towards Geometric Completion of Differential Systems by Points (p. 79-80)

Wenyuan Wu, Greg Reid and Oleg Golubitsky

Abstract Numerical Algebraic Geometry represents the irreducible components of algebraic varieties over C by certain points on their components. Such witness points are efficiently approximated by Numerical Homotopy Continuation methods, as the intersection of random linear varieties with the components. We outline challenges and progress for extending such ideas to systems of differential polynomials, where prolongation (differentiation) of the equations is required to yield existence criteria for their formal (power series) solutions. For numerical stability we marry Numerical Geometric Methods with the Geometric Prolongation Methods of Cartan and Kuranishi from the classical (jet) geometry of differential equations. Several new ideas are described in this article, yielding witness point versions of fundamental operations in Jet geometry which depend on embedding Jet Space (the arena of traditional differential algebra) into a larger space (that includes as a subset its tangent bundle). The first new idea is to replace differentiation (prolongation) of equations by geometric lifting of witness Jet points. In this process, witness Jet points and the tangent spaces of a jet variety at these points, which characterize prolongations, are computed by the tools of Numerical Algebraic Geometry and Numerical Linear Algebra. Unlike other approaches our geometric lifting technique can characterize projections without constructing an explicit algebraic equational representation.We first embed a given system in a larger space. Then using a construction of Bates et al., appropriate random linear slices cut out points, characterizing singular solutions of the differential system.

3.1 Introduction

3.1.1 Historical Background

Exact commutative algebra is concerned with commutative rings and their associated modules, rings and ideals. It is a foundation for algebraic geometry for polynomial rings amongst other areas. In our case, commutative algebra is a fundamental constituent of differential algebra for differential polynomial rings. Our paper is part of a collection that focuses on the rapidly evolving theory and algorithms for approximate generalizations of commutative algebra. The generalizations are nontrivial and promise to dramatically widen the scope and applications of the area of traditional exact commutative algebra.

Although the study of systems of differential polynomials (i.e. polynomially nonlinear PDE) is more complicated than algebraic systems of polynomials, historically key algorithmic concepts in commutative algebra, often arose initially for PDE. For example, differential elimination methods, arose first in the late 1800’s. In particular the classical methods of Riquier 1910 [18] and Tresse 1894 [30] for reducing systems of PDE to certain passive forms can in hindsight be regarded to implicitly contain versions of Buchberger’s Algorithm. However, the full potency and development of the theory had to await Buchberger’s work 1965 [5]. Indeed there is a well known isomorphism between polynomials and constant coefficient linear homogeneous PDE, which may be interpreted as mapping indeterminates to differential operators. Thus multiplication by a monomial maps to differentiation and reduction maps to elimination. Hence the Gröbner Basis algorithm is equivalent to a differential elimination method for such linear PDE. Further the Hilbert Function, gives the degree of generality of formal power series solutions of such PDE under this mapping.