Topics in Resultants and Implicitization

Abstract ii
Arknowletlgments iv
List of Illustrations xi
1 Introduction 1
1.1 Motivation ................................. 1
1.2 Background ................................ ‘2
1.2.1 Resultants ............................. '2
1.2.2 lmplicitization ........................... 4
1.3 Outline and Main Results ........................ 6
I Resultants 9
2 Univariate Resultants 10
2.1 The Sylvester and Bézout Resultants .................. 11
22.2 Exact Division by Truncated Formal Power Series ........... 1-1
2.3 The Transformation Matrix From Sylvester to Bézout ......... 15
2-1 Fast Computation of the Entries of the Bézout Resultant ....... 16
2.5 Computational Complexity ........................ ‘20
2.6 Hybrid Resultants ............................. 21
2.7 Non-Homogeneous Bézout Matrices ................... ‘24
3 Bivariate Tensor Product Resultants 29
3.1 The Three Dixon Resultants ....................... 31
3.1.1 The Sylvester Resultant Syl ( f. g, h) ...............
3.1.2 The Cayley Resultant Cay(f.g. h) ................
3.1.3 The Mixed Cayley-Sylvester Resultant 111111 f. g. h) ......
3.1.4 Notation ..............................
3.1.5 Review and Preview .......................
The Transformation from Syl(f. g. h) to .lfi;r(f. g, h) .........
3.2.1 The Conversion Matrix G(j'.g. h)‘ ................
3.2.2 The Entries of C(f. g. h)‘ ....................
3.2.3 Properties of G(f. g. h)’ .....................
The Transformation from S gl( f. g. h) to Cay/(f. g. h) .........
3.3.1 The Conversion Matrix F(f. g. h) ................
3.3.2 The Entries of F(f.g. h) .....................
3.3.3 Properties of F(f. g. h) ......................
The Transformation from .lli1'(f. g, h) to Cay”, g, h) .........
3.4.1 The Conversion Matrix E(f.g. h) ................
3.4.2 The Entries of E(f.g. h) .....................
3.4.3 Properties of E(f. g, h) ......................
The Block Structure of the Three Dixon Resultants ..........
3.5.1 The Block Structure of Sgl(f.g, h) ...............
3.5.2 The Block Structure of .lli.::(f.g. h) ...............
3.5.3 The Block Structure of Cay/(f. g. h) ...............
Convolution Identities ..........................
3.6.1 Interleaving in S.‘ and .\[,_. ....................
3.6.2 The Convolution Identities ....................
Fast Computation of the Entries of Cay(f,g. h) ............
Computational Complexity ........................
Hybrids of the Three Dixon Resultants .................
3.9.1 Hybrids of the Sylvester and Cayley Resultants ........
3.9.2 Hybrids of the Sylvester and Mixed Cayley-Sylvester Resultants 75
3.9.3 Hybrids of the Mixed Cayley-Sylvester and Cayley Resultants
4 Sparse Resultants
4.1 Construction of Sylvester A-resultants .................
4.1.1 Bi-degree Sylvester Resultant Matrices .............
4.1.2 Rectangular Corner Cut Sylvester A-resultant Matrices . . . .
4.2 Sylvester A-resultants ..........................
4.2.1 Only the Upper Right Corner is Cut Off ............
4.2.2 Cut Off One More Corner: The Upper Left Corner ......
4.2.3 Cut Off One. More Corner: The Lower Right Corner ......
4.2.4 Cut Off All Four Corners .....................
4.3 Remarks on Sylvester A-resultants ...................
4.4 Construction of Dixon A-resultants ...................
4.4.1 Bi-degree Dixon Resultant Matrices ...............
4.4.2 Corner Cut Dixon .A-resultant Matrices .............
4.5 Dixon A-resultants ............................
4.6 Comparison of the Sylvester and Dixon A-resultants ..........
4.7 Implicitization by Sylvester A-resultants ................
4.8 Implicitization by Dixon A-resultants ..................
4.8.1 Hirzebruch Surfaces ........................
4.8.2 Diamond Cyclides .........................
4.8.3 One-Horned Cyclides .......................
4.8.4 Hexagons .............................
4.8.5 Pentagons .............................
II Implicitization
5 Implicitization Using Moving Curves 122
5.1 The Method of Moving Curves ...................... 124
5.2 lmplicitizing Curves Using Moving Lines ................ 1'28
5.2.1 Even Degree Rational Curves .................. 128
5.2.2 Odd Degree Rational Curves ................... 132
5.2.3 .-\nti-.-\nnihilation by u—Basis .................. 133
5.3 lrnplicitizing Curves Using Moving Conics ............... 135
5.3.1 Moving Line and Moving Conic Coefficient Matrices ...... 136
5.3.2 |.\IL| Factors lMCwI ....................... 138
5.3.3 Moving Conics and Curves of Odd Degrees ........... 143
6 Implicitization Using Moving Surfaces 145
6.1 The Method of Moving Surfaces ..................... 147'
6.2 Implicitizing Surfaces Using Moving Planes ............... 151
6.3 Resultants and Syzygies ......................... 157
6.3.1 Triangular Polynomials ...................... 158
6.3.2 Tensor Product Polynomials ................... 159
6.4 Implicitizing Tensor Product Surfaces Using Moving Quadrics . . . . 161
6.4.1 Moving Plane and Moving Quadric Coefficient Matrices . . . . 161
6.4.2 The Validity of Moving Quadrics for Tensor Product Surfaces 162
6.5 Implicitizing Triangular Surfaces Using Moving Quadrics ....... 167
6.5.1 Moving Plane and Moving Quadric Coefficient Matrices . . . . 167
6.5.2 The Method of Moving Quadrics for Triangular Surfaces . . . 169
6.5.3 The Validity of Moving Quadrics for Triangular Surfaces . . . 170
III Open Questions 176
7 Open Questions 177
7.1 Classical Resultants ............................ 177
7.2 Sparse Resultants ............................. 179
7.3 Moving Quadrirs ............................. 181
Bibliography 185