| 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[x1,...,xn]
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 Il. Solving equations,
i.e. finding V(I), is now done iteratively: First
find V(In-1); substitute each solution
into a Groebner basis for In-2 to find
V(In-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
kn. Colon ideals, difference of
varieties. More on elimination theory:
V(Il) is the Zariski closure of the
projection of V(I).
|
| 3/11 | 4.4: Proof that V(Il)
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.
|