Table of contents for Using algebraic geometry / David A. Cox, John Little, Donal O'Shea.


Bibliographic record and links to related information available from the Library of Congress catalog. Note: Electronic data is machine generated. May be incomplete or contain other coding.


Counter





Preface to the Second Edition                                     v

Preface to the First Edition                                         vii

1 Introduction                                                    1
   1  Polynomials and Ideals . . . . . . . . . . . . . . . . . . . . . . .  1
   2   Monomial Orders and Polynomial Division .............. .  6
   3   Grobner Bases ........................          .........    13
   4   Affine Varieties ...........         . .............. 19

2 Solving Polynomial Equations                                       26
   1  Solving Polynomial Systems by Elimination . . . . . . . . ....  26
   2   Finite-Dimensional Algebras ....................... . 37
   3   Grobner Basis Conversion ................... .. 49
   4   Solving Equations via Eigenvalues and Eigenvectors ....... . 56
   5   Real Root Location and Isolation.  ................ .. . .. 69

3  Resultants                                                        77
    1 The Resultant of Two Polynomials . . . . . . . . . . . . ..... 77
    2  Multipolynomial Resultants ......................... 84
    3  Properties of Resultants ...........................     95
    4  Computing Resultants ............................ 102
    5   Solving Equations via Resultants .................. 114
    6   Solving Equations via Eigenvalues and Eigenvectors ....... . 128

4   Computation in Local Rings                                      137
    1  Local Rings .................. ........... 137
    2  Multiplicities and Milnor Numbers ................. 145
    3  Term Orders and Division in Local Rings ................ 158
    4   Standard Bases in Local Rings ....................... 174
    5   Applications of Standard Bases ................... 180




5  Modules                                                        189
   1   Modules over Rings . . . . .... .. . . . . . .  ... . . . 189
   2   Monomial Orders and Grobner Bases for Modules .......... . 207
   3   Computing Syzygies ...... . .     ... ....   . ..... . . 222
   4   Modules over Local Rings .....................234

6  Free Resolutions                                               247
    1  Presentations and Resolutions of Modules . . . . . . . . .... . 247
   2   Hilbert's Syzygy Theorem .....................258
   3   Graded Resolutions ... .  ........    ...... . .... . . 266
   4   Hilbert Polynomials and Geometric Applications ......... 280

7  Polytopes, Resultants, and Equations                           305
    1  Geometry of Polytopes . . . . . . . . . . . . . . ..... . . . . .. 305
   2   Sparse Resultants ..........................313
   3   Toric Varieties ..........      . . . . . . . . . . . . ....... . 322
   4   Minkowski Sums and Mixed Volumes ...............332
   5   Berstein's Theorem  . . . . . . . . . . . . . . ..........342
   6   Computing Resultants and Solving Equations ......... .357

8  Polyhedral Regions and Polynomials                             376
   1   Integer Programming ........................376
   2   Integer Programming and Combinatorics .............. 392
   3   Multivariate Polynomial Splines ..................405
   4   The Grobner Fan of an Ideal . . . . . . . . . . . . . .......426
   5   The Grobner Walk  . . . . . . . . . . . . .............436

9  Algebraic Coding Theory                                        451
     1 Finite Fields . . . . . . . . . . . . . . . . . . . .......... ...451
    2  Error-Correcting Codes ..... ......... . . .        ... .459
    3  Cyclic Codes ......    .... . . . . . . .............. . . 468
    4  Reed-Solomon Decoding Algorithms .................       . 480

10 The Berlekamp-Massey-Sakata Decoding Algorithm                 494
    1  Codes from Order Domains ..................           . .. 494
    2  The Overall Structure of the BMS Algorithm  ............ . 508
    3  The Details of the BMS Algorithm ................. 522

 References                                                       533

 Index                                                            547





Library of Congress Subject Headings for this publication: Geometry, Algebraic