| Day | Topics | Sections in IVA |
|---|---|---|
| Mon. 1/26 | The polynomial ring in one variable: Algbebraic structure |
IVA 1.5, H. Ch. 4 |
| Wed. 1/28 | The polynomial ring in one variable: Algorithms: Division, Euclidean, Radical Geometry: Roots of a univariate polynomial. |
IVA 1.5, H. Ch. 4 |
| Mon. 2/2 | Ideals in polynomial rings. Varieties: Solutions of a ideals. |
IVA 1.1-2, 4 |
| Wed. 2/4 | Parametrization and Implicitization. | I.2,3. II.1 |
| Wed. 2/11 | Monomial Orderings Division in multivariate polynomial rings. |
II.2, 3 |
| Fri. 2/13 | A menagerie of monomial orderings. Division Algorithm in Magma. |
II.2, 3 poly_div.mg |
| Mon. 2/16 | Dickson's Lemma. | II.4 |
| Wed. 2/18 | Hilbert basis theorem, Grobner basis. | II.5-8 |
| Mon. 2/23 | S-polynomials and Grobner basis.
Buchberger's algorithm. |
II.6-8 |
| Wed. 2/25 | Rings, ideals, prime and maximal ideals Homomorphisms, quotient rings. |
H Ch. 6 A Ch. 2 |
| Mon. 3/2 | Unique factorization domains. | A Ch. 2 |
| Wed. 3/4 | Localization, rings of fractions. | A Ch. 2 |
| Mon. 3/9 | R a UFD implies R[x] a UFD. | A Ch. 2 |
| Wed. 3/11 | Modules over a ring. | A Ch. 4.1,2,3,7 (just bits of each) |
| Mon. 3/16 | Presentations of a module and exact sequences. | A Ch. 4.1,2,3,7 (just bits of each) |
| Wed. 3/18 | The syzygy module for monomial ideals. Relation to reduction by G |
II.9 |
| Mon. 3/23 | Sylvester matrix, resultants. | III.5,6 |
| Wed. 3/25 | Proof of the Extension theorem. | III.5,6 |
| Mon. 4/6 | Implicitization of parametrically defined varieties. |
III.3 |
| Wed. 4/8 | Radical ideals and Hilbert's Nullstellensatz.
Zariski closure. |
IV.1,2 |
| Mon. 4/13 | The algebra-geometry dictionary. Sums, products and intersections of ideals. | IV.2,3. |
| Wed. 4/15 | Computation of the intersection of two ideals.
Proof of the closure theorem. Ideal quotients, and the geometric interpretation. | IV.3,4. |
| Mon. 4/20 | Irreducible varieties and minimal decomposition. | IV.5,6 |
| Wed. 4/22 | Polynomial maps on varieties. The coordinate ring of a variety. |
V.1-4 |
| Mon. 4/27 | Computing in the coordinate ring of a variety. Finite varieties. |
V.3 |
| Wed. 4/29 | Cone varieties, homogeneous ideals, the cone over a variety. | VIII.1-4x. |
| Mon. 5/4 | Projective geometry. | VIII.1-4. |
| Wed. 5/6 | Projective geometry and parametrized varieties. | VIII.1-4. |
| Mon. 5/11 | No class: discuss projects with me. | . |
| Wed. 5/13 | No class: discuss projects with me. | . |