# Readings

When you click the Amazon logo to the left of any citation and purchase the book (or other media) from Amazon.com, MIT OpenCourseWare will receive up to 10% of this purchase and any other purchases you make during that visit. This will not increase the cost of your purchase. Links provided are to the US Amazon site, but you can also support OCW through Amazon sites in other regions. Learn more. |

The topics follow the order of the required textbook:

Cox, David, John Little, and Donal O'Shea. *Ideals, Varieties, and Algorithms*. 3rd ed. Undergraduate Texts in Mathematics. New York, NY: Springer, 2007. ISBN: 9780387356518.

Portions of the book are online .

SES # | TOPICS | READINGS |
---|---|---|

1 | Polynomials and affine space, affine varieties | Sections 1-1 and 1-2 |

2 | Parameterizations of affine varieties, ideals | Sections 1-3 and 1-4 |

3 | Polynomials of one variable, orderings on the monomials in k[x,...,_{1}x]_{n} |
Sections 1-5, 2-1, and 2-2 |

4 | A division algorithm in k[x,...,_{1}x], monomial ideals and Dickson's lemma_{n} |
Sections 2-3 and 2-4 |

5 | The Hilbert basis theorem and Groebner bases, properties of Groebner bases | Sections 2-5 and 2-6 |

6 | Buchberger's algorithm, first applications of Groebner bases | Sections 2-7 and 2-8 |

7 | The elimination and extension theorems, the geometry of elimination | Sections 3-1 and 3-2 |

8 | Implicitization, singular points and envelopes | Sections 3-3 and 3-4 |

9 | Unique factorization and resultants | Section 3-5 |

10 | Resultants and the extension theorem, the nullstellensatz | Sections 3-6 and 4-1 |

11 | Radical ideals and the ideal-variety correspondence, sums, products, and intersections of ideal | Sections 4-2 and 4-3 |

12 | Zariski closure and quotients of ideals, irreducible varieties and prime ideals | Sections 4-4 and 4-5 |

13 | Decomposition of a variety into irreducibles, polynomial mappings | Sections 4-6 and 5-1 |

14 | Quotients of polynomials R, algorithmic computations in k[x,...,_{1}x]/_{n}I |
Sections 5-2 and 5-3 |

15 | The coordinate ring of an affine variety, rational functions on a variety | Sections 5-4 and 5-5 |

16 | Proof of the Closure theory, geometric description of robots, the forward kinematics problem | Sections 5-6, 6-1, and 6-2 |

17 | The inverse kinematic problem and motion planning, automatic geometric theorem proving | Sections 6-3 and 6-4 |

18 | Wu's method, symmetric polynomials | Sections 6-5 and 7-1 |

19 | Finite matrix groups and rings of invariants, generators for the ring of invariants | Sections 7-2 and 7-3 |

20 | Relations among generators and the geometry of orbits, the projective plane, projective space and projective varieties | Sections 7-4, 8-1, and 8-2 |

21 | The projective algebra-geometry dictionary, the projective closure of an affine variety | Sections 8-3 and 8-4 |

22 | Projective elimination theory | Section 8-5 |

23 | The geometry of quadric hypersurfaces, the variety of a monomial ideal | Sections 8-6 and 9-1 |

24 | The complement of a monomial ideal, the Hilbert function and the dimension of a variety | Sections 9-2 and 9-3 |

25 | Elementary properties of dimension, dimension and algebraic independence | Sections 9-4 and 9-5 |

26 | Dimension and nonsingularity, the tangent cone | Sections 9-6 and 9-7 |