Dickenstein / Schreyer / Sommese Algorithms in Algebraic Geometry

1. Auflage 2010
ISBN: 978-0-387-75155-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 146, 162 Seiten

Reihe: The IMA Volumes in Mathematics and its Applications

In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its Applications on September 2006 is one tangible indication of the interest. This volume of articles captures some of the spirit of the IMA workshop.

Autoren/Hrsg.

Dickenstein, Alicia

Schreyer, Frank-Olaf

Sommese, Andrew J.

Inhaltsverzeichnis

1;FOREWORD;6
2;PREFACE;7
3;Table of Contents
;10
4;APPLICATION OF A NUMERICAL VERSION OF TERRACINI'S LEMMA FOR SECANTS AND JOINS
;11
4.1;1. Introduction;11
4.2;2. Background;13
4.2.1;2.1. Homotopy continuation;13
4.2.2;2.2. Singular value decomposition;14
4.2.3;2.3. Terracini's lemma;15
4.2.4;2.4. Bertini software package;16
4.3;3. Five illustrative examples;16
4.3.1;3.1. Secant variety of the Veronese surface in P5
;17
4.3.2;3.2. Two more examples;19
4.4;4. Further computational experiments;20
4.4.1;4.1. Computations on systems which are close together;20
4.4.2;4.2. Determining equations from a generic point;21
4.5;5. Conclusions;22
4.6;REFERENCES;23
5;ON THE SHARPNESS OF FEWNOMIAL BOUNDS AND THE NUMBER OF COMPONENTS OF FEWNOMIAL HYPERSURFACES
;25
5.1;1. Introduction;25
5.1.1;1.1. A lower bound for fewnomial systems;26
5.1.2;1.2. An upper bound for fewnomial hypersurfaces;27
5.2;REFERENCES;29
6;INTERSECTIONS OF SCHUBERT VARIETIES AND OTHER PERMUTATION ARRAY SCHEMES
;31
6.1;1. Introduction;31
6.2;2. The flag manifold and Schubert varieties;33
6.3;3. Permutation arrays;36
6.4;4. Permutation array varieties/schemes and their pathologies;41
6.5;5. Intersecting Schubert varieties;46
6.5.1;5.1. Permutation array algorithm;47
6.5.2;5.2. Algorithmic complexity;51
6.6;6. The key example: triple intersections;53
6.7;7. Monodromy and Galois groups;58
6.8;8. Acknowledgments;62
6.9;REFERENCES;63
7;EFFICIENT INVERSION OF RATIONAL MAPS OVER FINITE FIELDS
;65
7.1;1. Introduction;65
7.1.1;1.1. Outline of our approach;66
7.2;2. Notions and notations;67
7.2.1;2.1. Data structures;68
7.2.2;2.2. The algorithmic model;68
7.2.3;2.3. Cost of the basic operations;69
7.3;3. Geometric solutions;69
7.3.1;3.1. Algorithmic aspects of the computation of a geometric solution
;71
7.4;4. Preparation of the input data;72
7.4.1;4.1. The graph of the mapping F
;72
7.4.2;4.2. Random choices;73
7.5;5. The algorithm;76
7.5.1;5.1. The computation of the polynomial ms;78
7.5.2;5.2. A geometric solution of C;81
7.5.3;5.3. Computation of the points of F-1(y((O)) nFqn
;83
7.6;6. Conclusions;85
7.7;REFERENCES;86
8;HIGHER-ORDER DEFLATION FOR POLYNOMIAL SYSTEMS WITH ISOLATED SINGULAR SOLUTIONS
;89
8.1;1. Introduction;89
8.2;2. Statement of the main theorem & algorithms;91
8.3;3. Multiplicity structure;93
8.3.1;3.1. Standard bases;94
8.3.2;3.2. Dual space of differential functionals;94
8.3.3;3.3. Dual bases versus standard bases;95
8.4;4. Computing the multiplicity structure;96
8.4.1;4.1. The Dayton-Zeng algorithm;96
8.4.2;4.2. The Stetter-Thallinger algorithm;97
8.5;5. Proofs and algorithmic details;99
8.5.1;5.1. First-order deflation;99
8.5.2;5.2. Higher-order deflation with fixed multipliers;100
8.5.3;5.3. Indeterminate multipliers;103
8.6;6. Computational experiments;103
8.6.1;6.1. A first example;103
8.6.2;6.2. A larger example;104
8.7;7. Conclusion;105
8.8;REFERENCES;105
9;POLARS OF REAL SINGULAR PLANE CURVES;109
9.1;1. Introduction;109
9.2;2. Polar varieties;110
9.2.1;2.1. Classical polar varieties;110
9.2.2;2.2. Reciprocal polar varieties;113
9.3;3. Polar varieties of real singular curves;116
9.3.1;3.1. Classical polar varieties of real singular affine curves;117
9.3.2;3.2. Reciprocal polar varieties of affine real singular curves;119
9.4;REFERENCES;125
10;SEMIDEFINITE REPRESENTATION OF THE K-ELLIPSE;127
10.1;1. Introduction;127
10.2;2. Derivation of the matrix representation;130
10.3;3. More pictures and some semidefinite aspects;132
10.4;4. Generalizations;137
10.4.1;4.1. Weighted k-ellipse;137
10.4.2;4.2. k-Ellipsoids;138
10.5;5. Open questions and further research;139
10.6;REFERENCES;142
11;SOLVING POLYNOMIAL SYSTEMS EQUATION BY EQUATION
;143
11.1;1. Introduction;143
11.2;2. A numerical irreducible decomposition;144
11.2.1;2.1. An illustrative example;144
11.2.2;2.2. Witness sets;145
11.2.3;2.3. Geometric resolutions and triangular representations;146
11.2.4;2.4. Embeddings and cascades of homotopies;148
11.3;3. Application of diagonal homotopies;149
11.3.1;3.1. Symbols used in the algorithms;149
11.3.2;3.2. Solving subsystem by subsystem;151
11.3.3;3.3. Solving equation by equation;154
11.3.4;3.4. Seeking only nonsingular solutions;156
11.4;4. Computational experiments;157
11.4.1;4.1. An illustrative example;157
11.4.2;4.2. Adjacent minors of a general 2-by-9 matrix;157
11.4.3;4.3. A general 6-by-6 eigenvalue problem;159
11.5;5. Conclusions;160
11.6;REFERENCES;161
12;LIST OF WORKSHOP PARTICIPANTS;163
13;IMA VOLUMES;171