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[x1,…,xn] | Sections 1-5, 2-1, and 2-2 |
| 4 | A division algorithm in k[x1,…,xn], monomial ideals and Dickson’s lemma | 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[x1,…,xn]/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 |
