## 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.

```

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