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 |