Spring 2017

**Instructor:** Brendon Rhoades

**Instructor's Email:** bprhoades (at) math.ucsd.edu

**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.