Math 202C: Applied Algebra III
Spring 2017

Instructor's Office: 7250 APM
Instructor's Office Hours: 4:00-5:00pm MWF
Lecture Time: 3:00-3:50pm MWF
Lecture Room: 7421 APM

Lectures:

4/3: Administrivia. Chapter 1. The polynomial ring k[x_1, ... , x_n]. Ideals. Affine n-space k^n. Varieties.
4/5: Chapter 2. Review of polynomial divison in k[x]. Monomial orders on k[x_1, ... , x_n]. Leading terms, coefficients, monomials, and multidegree. Polynomial long division on k[x_1, ... , x_n]. Failure of uniqueness of remainder. Monomial ideals. Statement of Dickson's Lemma.
4/7: Chapter 2. Proof of Dickson's Lemma. Application to monomial orders. Definition of Groebner bases. Groebner bases exist. Groebner bases are bases. Hilbert's Basis Theorems. Equivalent formulations of Noetherian rings. Ideal Membership Problem and Groebner bases.
4/10: Chapter 2. Uniqueness of Remainder for Groebner bases. S-polynomials. Cancellation Lemma. Statement and proof of Buchberger's Criterion for deciding when a basis for an ideal is Groebner.
4/12: Chapter 2. Buchberger's Algorithm for turning a basis for an ideal I into a Groebner basis. Example. Proof of Buchberger's algorithm. Minimal Groebner bases; converting a Groebner basis into a minimal Groebner basis. Reduced Groebner bases; converting a minimal Groebner basis into a reduced Groebner bases. Fixing a monomial order, every ideal I in k[x_1, ... , x_n] has a unique reduced Groebner basis.
4/14: Chapter 2. The standard monomial basis for k[x_1, ... , x_n]/I with respect to a fixed term order. Chapter 3. Varieties, Zariski topology on affine space. Example of finding a variety. Elimination ideals. Statement and proof of the Elimination Theorem. Statement of the Extension Theorem.
4/17: Chapter 3. Failure of the Extension Theorem over R. Projections. Geometric version of the Extension Theorem. Partial and extendable solutions. Statement of the Closure Theorem. Failure of the Closure Theorem over R. Proof of Part 2 of the Closure Theorem in codimension 1, assuming the Extension Theorem and Part 1.
4/19: Chapter 3. Zariski constructible sets and the Closure Theorem. Polynomial parameterization. The implicitization problem. Implicitization via elimination ideals. The Implicitization Theorem over infinite fields. Units and irreducibles in k[x_1, ... , x_n]. k[x_1, ... , x_n] is a Unique Factorization Domain.
4/21: Chapter 3. Af + Bg = 0 criterion for polynomials f, g in k[x] to have a common factor. Sylvester matrix Syl(f,g,x). Resultant Res(f,g,x) = det Syl(f,g,x). f, g have a common factor iff Res(f,g,x) = 0. Res(f,g,x) is a polynomial in the coefficients of f,g. If f, g are in k[x_1, ... , x_n] then Syl(f,g,x_1) lies in the intersection of < f, g > with the polynomial ring k[x_2, ... , x_n] (Cramer's Rule).
4/24: Chapter 3. Using resultants to prove the Extension Theorem. Chapter 4. Statement and proof of the Weak Nullstellensatz. Failure of the Weak Nullstellensatz over a field which is not algebraically closed.
4/26: Chapter 4. Radical ideals. Statement and proof of Hilbert's Nullstellensatz. Failure of the Nullstellensatz over a field which is not algebraically closed. Ideal-Variety correspondence. Sums, products, and intersections of ideals.
4/28: Chapter 4. Sums, products, and intersections of ideals; correspondence with varieties. Computing bases of sums, products, and intersections of ideals. Prime ideals. Irreducible varieties. The irreducible decomposition of a variety.
5/1: Chapter 5. Polynomial (aka regular) maps between varieties V and W. Variety isomorphism. Examples. The coordinate ring k[V] of a variety V. A polynomial map from V to W induces a ring homomorphism from k[W] to k[V] fixing k.
5/3: Chapter 7. The action of the matrix group GL_n(k) (as well as its subgroups) on the polynomial ring k[x_1, ... , x_n]. Invariant rings. Basic properties of invariant rings. Examples of invariant rings.
5/8: Chapter 7. The Reynolds operator on k[x_1, ... , x_n]. Finite generation as a k-algebra. Not every subalgebra of a finitely generated k-algebra is finitely generated. Noether's Theorem: If char(k) = 0 and G is a finite subgroup of GL_n(k), the invariant space k[x_1, ... , x_n]^G is finitely generated as a k-algebra (and is generated by R_G(x), where x ranges over all monomials of total degree at most |G|. Proof of Noether's Theorem with symmetric functions.
5/10: Graded vector spaces. Hilbert series of a graded vector space. Examples. Hilbert series of invariant spaces k[x_1,...,x_n]^G. Molien's Theorem. Proof of Molien's Theorem using the Reynolds Operator.
5/12: Reflections in GL(V) for a Euclidean space V. Reflection groups. Examples: S_n, signed permutation matrices, type D_n, dihedral groups. Invariant rings of reflection groups; some examples. Uniqueness of degrees using pole analysis of Hilbert series.
5/15: More on reflection groups. If G is a finite subgroup of GL_n(R) with R[x_1, ... , x_n]^G polynomial, then R[x_1, ... , x_n]^G has n algebraically independent homogeneous generators. The product of the degrees of these generators is the order |G| of G. Examples. Statement that the sum of the degrees is n plus the number of reflections. Examples: Types A,B/C,I.
5/17: Proof of the "degree sum" result using Molien theory. Review of Jacobians and the Jacobian Criterion. The Shephard-Todd Theorem: If a finite subgroup G of GL_n(R) has invariant ring polynomial, then G is a reflection group. Proof of Shephard-Todd assuming the Chevallay Theorem.

Homework Assignments:

Homework 1, due 4/12/2017.
Homework 2, due 4/26/2017.
Homework 3, due 5/10/2017.