Date | Material covered |
---|---|
1/8 | Introduction by M. Lucianovic |
1/10 | 1.1, 1.2: polynomials and varieties |
1/15 | 1.4: Ideals and (something about) their relation to varieties. |
1/17 | 1.4,1.5: Polynomials in one variable. |
1/22 | 1.5: Euclid's algorithm. Algorithm for
deciding whether a polynomial f is an element of
<f_1, f_2, ..., f_s>. Started the study of
more variables (from 2.2). Introduced monomial
orderings in order to talk about "leading terms" in
more variables. Proved that lex is a monomial
ordering. |
1/24 | 2.2: More on monomial orderings. 2.3: the division algorithm. |
1/29 | 2.3: the division algorithm continued (we gave a mathematical proof that it works). Started 2.4: Dickson's lemma. |
1/31 | 2.4: Proof of Dickson's lemma. An easier criterion for monomial orders. Proof of Hilbert's basis theorem. |
2/5 | 2.5: Consequences of Hilbert's basis theorem: More on the correspondence between ideals and varieties. k[x_{1},...,x_{n}] satisfies the ascending chain condition. Introduction to Groebner bases. Uniqueness of remainders. Example and non-example. Introduction to Buchberger's criterion. |
2/7 | 2.5, 2.6: Formulate and prove Buchberger's criterion. |
2/12 | 2.7: Buchberger's algorithm (turn an arbitrary basis into a Groebner basis). Proof uses the "ascending chain condition" crucially. Minimal and reduced Groebner bases. Uniqueness of reduced Groebner bases. Relation to Reduced row echelon form for matrices. |
2/14 | 2.8,3.1: Elimination theory: Elimination ideals I_{l}. Solving equations, i.e. finding V(I), is now done iteratively: First find V(I_{n-1}); substitute each solution into a Groebner basis for I_{n-2} to find V(I_{n-2}); etc. Theoretical basis is the "elimination theorem", which we also proved and gave an example. |
2/19 | 3.1, 3.2, 4.1: The extension theorem. An example from the book, as illustration of the elimination and extension theorems. Another example to show that doing Groebner basis calculations by hand can be bad for the rain forest. Then we returned to the variety-ideal correspondence in chapter 4. I stated "Hilbert's Nullstellensatz". Note: For now, we'll skip the remainder of chapter 3 (including the proof of the extension theorem in 3.6). |
2/21 | 4.1, 4.2: Hilbert's Nullstellensatz. Radical ideals. The radical of an ideal. Examples. Special case where V(I) is empty is the "Weak Nulstellensatz" (higher dimensional analogue of the fundamental theorem of algebra). I started the proof of the Weak Nulstellensatz. |
2/26 | 4.1, 4.2: Proof of weak and strong Nulstellensatz. "Radical Membership" algorithm. Operations on ideals (from 4.3). |
2/28 | 4.3, 4.5: Intersection of ideals (and algorithm for computing generators). Prime ideals and irreducible varieties. |
3/4 | 4.5, 4.6: Prime ideals and irreducible varieties. Decompositions of varieties into irreducibles: Existence, Uniqueness, examples. Maximal ideals. |
3/6 | 4.5, 4.4: Maximal ideals correspond to points in varieties (when the ground field is algebraically closed). Zariski closure of subsets of k^{n}. Colon ideals, difference of varieties. More on elimination theory: V(I_{l}) is the Zariski closure of the projection of V(I). |
3/11 | 4.4: Proof that V(I_{l}) is the Zariski closure of the projection of V(I). Then I jumped to Chapter 9 and defined the dimension of a variety as the degree of the Hilbert polynomial, motivated by some examples. |
3/13 | 9.2,9.3: Proof that the Hilbert function is polynomial for large s. |