In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric compuation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry.
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 18-22, 2006 at the University of Minnesota is one tangible indication of the interest. One hundred ten participants from eleven countries and twenty states came to listen to the many talks; discuss mathematics; and pursue collaborative work on the many faceted problems and the algorithms, both symbolic and numberic, that illuminate them.
This volume of articles captures some of the spirit of the IMA workshop.
Author(s): Alicia Dickenstein, Frank-Olaf Schreyer, Andrew J. Sommese
Series: The IMA Volumes in Mathematics and its Applications
Edition: 1
Publisher: Springer
Year: 2007
Language: English
Pages: 174
Cover......Page 1
The IMA Volumes in Mathematics and its Applications Volume 146......Page 2
Algorithms in Algebraic Geometry......Page 4
0387751548......Page 5
FOREWORD......Page 6
PREFACE......Page 8
Table of Contents
......Page 12
1. Introduction......Page 14
2.1. Homotopy continuation......Page 16
2.2. Singular value decomposition......Page 17
2.3. Terracini's lemma......Page 18
3. Five illustrative examples......Page 19
3.1. Secant variety of the Veronese surface in P5
......Page 20
3.2. Two more examples......Page 22
4.1. Computations on systems which are close together......Page 23
4.2. Determining equations from a generic point......Page 24
5. Conclusions......Page 25
REFERENCES......Page 26
1. Introduction......Page 28
1.1. A lower bound for fewnomial systems......Page 29
1.2. An upper bound for fewnomial hypersurfaces......Page 30
REFERENCES......Page 32
1. Introduction......Page 34
2. The flag manifold and Schubert varieties......Page 36
3. Permutation arrays......Page 39
4. Permutation array varieties/schemes and their pathologies......Page 44
5. Intersecting Schubert varieties......Page 49
5.1. Permutation array algorithm......Page 50
5.2. Algorithmic complexity......Page 54
6. The key example: triple intersections......Page 56
7. Monodromy and Galois groups......Page 61
8. Acknowledgments......Page 65
REFERENCES......Page 66
1. Introduction......Page 68
1.1. Outline of our approach......Page 69
2. Notions and notations......Page 70
2.2. The algorithmic model......Page 71
3. Geometric solutions......Page 72
3.1. Algorithmic aspects of the computation of a geometric solution
......Page 74
4.1. The graph of the mapping F
......Page 75
4.2. Random choices......Page 76
5. The algorithm......Page 79
5.1. The computation of the polynomial ms......Page 81
5.2. A geometric solution of C......Page 84
5.3. Computation of the points of F-1(y((O)) nFqn
......Page 86
6. Conclusions......Page 88
REFERENCES......Page 89
1. Introduction......Page 92
2. Statement of the main theorem & algorithms......Page 94
3. Multiplicity structure......Page 96
3.2. Dual space of differential functionals......Page 97
3.3. Dual bases versus standard bases......Page 98
4.1. The Dayton-Zeng algorithm......Page 99
4.2. The Stetter-Thallinger algorithm......Page 100
5.1. First-order deflation......Page 102
5.2. Higher-order deflation with fixed multipliers......Page 103
6.1. A first example......Page 106
6.2. A larger example......Page 107
REFERENCES......Page 108
1. Introduction......Page 112
2.1. Classical polar varieties......Page 113
2.2. Reciprocal polar varieties......Page 116
3. Polar varieties of real singular curves......Page 119
3.1. Classical polar varieties of real singular affine curves......Page 120
3.2. Reciprocal polar varieties of affine real singular curves......Page 122
REFERENCES......Page 128
1. Introduction......Page 130
2. Derivation of the matrix representation......Page 133
3. More pictures and some semidefinite aspects......Page 135
4.1. Weighted k-ellipse......Page 140
4.2. k-Ellipsoids......Page 141
5. Open questions and further research......Page 142
REFERENCES......Page 145
1. Introduction......Page 146
2.1. An illustrative example......Page 147
2.2. Witness sets......Page 148
2.3. Geometric resolutions and triangular representations......Page 149
2.4. Embeddings and cascades of homotopies......Page 151
3.1. Symbols used in the algorithms......Page 152
3.2. Solving subsystem by subsystem......Page 154
3.3. Solving equation by equation......Page 157
3.4. Seeking only nonsingular solutions......Page 159
4.2. Adjacent minors of a general 2-by-9 matrix......Page 160
4.3. A general 6-by-6 eigenvalue problem......Page 162
5. Conclusions......Page 163
REFERENCES......Page 164
LIST OF WORKSHOP PARTICIPANTS......Page 166
IMA VOLUMES......Page 174
