Math 202C: Applied Algebra III
Spring 2015

Instructor's Email: bprhoades (at) math.ucsd.edu
Instructor's Office: 7250 APM
Instructor's Office Hours: 10:00-11:00am MWF or by appointment
Lecture Time: 2:00-2:50pm MWF
Lecture Room: 5402 APM

Homework: 100%

Textbook: Ideals, Varieties, and Algorithms by Cox-Little-O'Shea

Lectures:

3/30: Chapter 1. Rings, polynomials, monomials, terms, coefficients, multidegree, total degree, affine n-space k^n, evaluation maps, the variety V(S) cut out by a set of polynomials S in k[x_1, ... , x_n], Zariski topology, ideals, the ideal I = < S > generated by a set S of ring elements.
4/1: Chapter 1. The ideal I(X) of a subset X of k^n. A basis of an ideal. Noetherian rings. Statement of Hilbert's Basis Theorem. The ring k[x]. The Division Algorithm. Section 2.2. Monomial orders.
4/3: Section 2.2. Lex order, grlex order, grevlex order. Monomial orders are extensions of the componentwise partial order. Leading terms, monomials, coefficients. Multidegree. Section 2.3. The division algorithm in k[x_1, ... , x_n] and its failure to have nice properties in general. Section 2.4. Monomial ideals. Dickson's Lemma.
4/6: Section 2.4. A translation invariant total order on monomials is a monomial order iff 0 is the minimum element. Section 2.5. Definition of Groebner bases. Existence of Groebner bases (Dickson's Lemma). Groebner bases are bases -- Hilbert's Basis Theorem. Section 2.6. Uniqueness properties of the remainder upon division by a Groebner basis. The ideal membership problem solution. S-polynomials. Statement of Buchberger's Criterion.
4/8: Section 2.6. Proof of Buchberger's Criterion. Section 2.7. Statement of Buchberger's Algorithm.
4/10: Section 2.7. Example of Buchberger's Algorithm. Proof of the correctness of Buchberger's Algorithm. Minimal and reduced Groebner bases. Every ideal in k[x_1, ... , x_n] has a unique reduced Groebner basis with respect to a given term order. Section 3.1. Elimination ideals. The Elimination Theorem for finding bases of elimination ideals.
4/13: Section 3.1. Proof of the Elimination Theorem. Example of using elimination theory to determine varieties. Partial solutions: extendable and not-extendable. Statement of the Extension Theorem. Example to show the Extension Theorem fails over not-algebraically closed fields.
4/15: Sections 3.1 and 3.2. More on geometry and the Extension Theorem. Statement of the Closure Theorem. Proof of half of the Closure Theorem in codimension 1.
4/17: Prof. Rhoades was in Las Vegas. Class cancelled.
4/20: Prof. Rhoades was in Las Vegas. Class cancelled.
4/22: Section 3.3. Polynomial implicitization. Section 3.5/6. Definition of the resultant Res(f, g, x_1).
4/23: Section 3.5/3.6. The resultant Res(f, g, x_1) lies in the first elimination ideal of < f, g >. Proof of the Extension Theorem. Section 4.1. Statement and proof of the Weak Nullstellensatz.
4/24: Section 4.1. Consistency algorithm for polynomial systems over algebraically closed fields. Statement and proof of Hilbert's Nullstellensatz. Section 4.2. Radical ideals. Ideal-variety correspondence. Ideal membership algorithm. Section 4.3. Ideal sum, product, and intersection. Translation to varieties. Ideal intersection algorithm.
4/27: Section 4.4. Zariski closure. Proof of (part 1 of) the Closure Theorem. Quotient ideals and difference of varieties. Section 4.5. Irreducible varieties and prime ideals. Zariski closures of parameterizations are irreducible.
4/29: Chapter 5. Regular (polynomial) maps between varieties. Coordinate rings. Functorial properties of coordinate rings. Isomorphism of varieties. Two varieties are isomorphic if and only if their coordinate rings are isomorphic. Examples: The line is isomorphic to the moment curve. The line is not isomorphic to x^2 = y^3.
4/30: Chapter 7. The action of GL_n(k) (and its subgroups) on k[x_1, ... , x_n]. Invariant rings. Examples.
5/1: Chapter 7. More examples of invariant rings. Hilbert series. The Hilbert series of invariant rings.
5/4: Chapter 7. The Reynolds operator R_G. Noether's Theorem on generation of invariant rings. Example of Noether's Theorem. Proof of Noether's Theorem.
5/6: Chapter 7. Given polynomials f_1, ... , f_r, how to decide if another polynomial f is a polynomial combination of the f_i, and how to express f as a polynomial combination of the f_i. Statement of Molien's Theorem. S_3-example.
5/8: Another example of Molien's Theorem. Proof of Molien's Theorem using the Reynolds operator.
5/13: If G is a finite subgroup of GL_n(k) such that k[x_1, ... , x_n]^G has algebraically independent homogeneous generators f_1, ... , f_r, then n = r. The degrees d_i = deg(f_i) are uniquely determined (independent of the choice of f_i). The product d_1 d_2 ... d_n of the degrees is |G|. Reflections and reflection groups. Irreducible reflection groups. Real irreducible reflection groups.

Homework Assignments:

Homework 1, Due 4/10/2015.
Homework 2, Due 4/22/2015.
Homework 3, Due 5/4/2015.
Homework 4. Not to be handed in.