| From Oil Fields to Hilbert Schemes | p. 1 |
| A Problem Arising in Industrial Mathematics | p. 5 |
| Border Bases | p. 10 |
| The Eigenvalue Method for Solving Polynomial Systems | p. 18 |
| Approximate Vanishing Ideals | p. 22 |
| Stable Order Ideals | p. 31 |
| Border Basis and Gröbner Basis Schemes | p. 40 |
| References | p. 53 |
| Numerical Decomposition of the Rank-Deficiency Set of a Matrix of Multivariate Polynomials | p. 55 |
| Background Material | p. 59 |
| Genericity and Randomness | p. 59 |
| The Numerical Irreducible Decomposition | p. 60 |
| Images of Algebraic Sets | p. 62 |
| Random Coordinate Patches on Grassmannians | p. 63 |
| Finding Rank-Dropping Sets | p. 65 |
| Generalizations | p. 67 |
| Applications | p. 69 |
| Support of a Module | p. 69 |
| Degeneracy Sets of the Differential of a Map | p. 70 |
| Singular Sets | p. 70 |
| Implementation Details and Computational Results | p. 71 |
| Singular Set for a Matrix | p. 71 |
| Singular Set for a Hessian Matrix | p. 71 |
| Singular Solutions for a Polynomial System | p. 72 |
| The Singular Set of the Reduction of an Algebraic Set | p. 73 |
| Equations Defining an Algebraic Set | p. 73 |
| Computing the Singular Set of the Reduction of an Algebraic Set | p. 75 |
| References | p. 75 |
| Towards Geometric Completion of Differential Systems by Points | p. 79 |
| Introduction | p. 80 |
| Historical Background | p. 80 |
| Exact Differential Elimination Algorithms | p. 81 |
| Outline of Paper | p. 82 |
| Zero Sets of PDE | p. 82 |
| Witness Sets of PDE | p. 83 |
| Witness Jet Points | p. 84 |
| Witness Tangent Space | p. 85 |
| Geometric Lifting and Singular Components | p. 86 |
| Determination of Singular Components of an ODE using Numerical Jet Geometry | p. 88 |
| Determination of Singular Components of a PDE System | p. 90 |
| Discussion | p. 94 |
| References | p. 95 |
| Geometric Involutive Bases and Applications to Approximate Commutative Algebra | p. 99 |
| Jet Spaces and Geometric Involutive Bases | p. 102 |
| Jet Geometry and Jet Space | p. 102 |
| Prolongation and Projection | p. 103 |
| The Symbol | p. 105 |
| Indices and Cartan Characters | p. 105 |
| The Cartan-Kuranishi Prolongation Theorem | p. 106 |
| Geometric Projected Involutive Bases and Nearby Systems | p. 107 |
| Geometric Projected Involutive Bases | p. 107 |
| Approximately Involutive Systems | p. 108 |
| Nearby Systems: Structure and Convergence | p. 110 |
| The Hilbert Function | p. 112 |
| Definition and Key Properties | p. 112 |
| Connection with Involutive Systems | p. 112 |
| A Motivational Example | p. 113 |
| Applications | p. 114 |
| Ideal Membership | p. 114 |
| Gröbner Bases for Polynomial Systems | p. 115 |
| Appendix | p. 119 |
| The SVD, $-Rank, and $-Rank | p. 119 |
| STLS | p. 120 |
| STLS-RREF | p. 121 |
| References | p. 123 |
| Regularization and Matrix Computation in Numerical Polynomial Algebra | p. 125 |
| Notation and preliminaries | p. 127 |
| Notation | p. 127 |
| Numerical rank and kernel | p. 128 |
| The linear and nonlinear least squares problems | p. 131 |
| Fonnulation of the approximate solution | p. 134 |
| The ill-posed problem and the pejorative manifold | p. 134 |
| The three-strikes principle for removing ill-posedness | p. 138 |
| Matrix computation arising in polynomial algebra | p. 142 |
| Approximate GCD | p. 142 |
| The multiplicity structure | p. 144 |
| Numerical elimination | p. 146 |
| Approximate irreducible factorization | p. 147 |
| A subspace strategy for efficient matrix computations | p. 150 |
| The closedness subspace for multiplicity matrices | p. 150 |
| The fewnomial subspace strategy for multivariate polynomials | p. 153 |
| Software development | p. 155 |
| References | p. 158 |
| Ideal Interpolation: Translations to and from Algebraic Geometry | p. 163 |
| Introduction | p. 163 |
| Ideal projectors | p. 164 |
| Pararnetrization | p. 165 |
| Multiplication operators | p. 167 |
| Duality | p. 168 |
| Hennite Projectors and Their Relatives | p. 170 |
| Perturbations of ideal projectors | p. 170 |
| Lagrange and curvilinear projectors | p. 171 |
| Limits of Lagrange and curvilinear projectors | p. 173 |
| Two problems | p. 176 |
| Existence of non-Hermite projectors | p. 177 |
| Description of non-Hermite projectors | p. 178 |
| Projectors in three variables | p. 179 |
| Nested Ideal Interpolation | p. 180 |
| Ideal restrictions | p. 181 |
| A conjecture of Tomas Sauer | p. 182 |
| Divided differences | p. 183 |
| Ideal decomposition | p. 383 |
| Error Formula | p. 185 |
| Loss of Haar | p. 187 |
| References | p. 190 |
| An Introduction to Regression and Errors in Variables from an Algebraic Viewpoint | p. 193 |
| Regression and the X -matrix | p. 193 |
| Orthogonal polynomials and the residual space | p. 196 |
| The fitted function and its variance | p. 198 |
| ôErrors in variablesö analysis of polynomial models | p. 199 |
| Comments | p. 201 |
| Acknowledgements | p. 202 |
| References | p. 202 |
| ApCoA = Embedding Commutative Algebra into Analysis | p. 205 |
| Introduction | p. 205 |
| Approximate Commutative Algebra | p. 206 |
| Empirical Data | p. 207 |
| Valid Results; Validity Checking of Results | p. 208 |
| Data → Result Mappings | p. 209 |
| Analytic View of Data→Result Mappings | p. 210 |
| Condition | p. 211 |
| Overdetermination | p. 213 |
| Syzygies | p. 214 |
| Singularities | p. 215 |
| Conclusions | p. 217 |
| References | p. 217 |
| Exact Certification in Global Polynomial Optimization Via Rationalizing Sums-Of-Squares | p. 219 |
| References | p. 225 |
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