# Grobner Bases and Algebraic Geometry, Spring 2008

### Content

- Varieties, ideals, and algorithms.
- Monomial orderings and the division algorithm.
- Dickson's lemma and the Hilbert basis theorem.
- Groebner bases and Buchberger's algorithm.
- Solving systems of polynomial equations using Groebner bases.
- Implicitization.
- Resultants.
- Hilbert's Nullstellensatz and radical ideals.
- Irreducible varieties and prime ideals.
- The prime(ary) decomposition of ideals.
- Quotient rings.
- Proving theorems in geometry.

#### Textbook

*Ideals, Varieties and Algorithms* by Cox, Little, O'Shea

#### Software

We will use Maple extensively for calculations and programming in this course. The university has a site license. Maple is installed on the PCs and MACs in the assignment lab, the CECM lab, university open labs and the library. Maple is available from the microcomputer store for about $180.

#### Handouts

Course info sheet (.txt)

Maple worksheet showing how to use Maple (.txt)

Maple worksheet (February 7th) for how to use Groebner package

Maple worksheet on implicititization (February 14th)

Resultant worksheet (February 21st)

Maple worksheet on the radical of an ideal (February 28th)

Ideal intersection worksheet (March 4th)

Ideal quotient worksheet (March 6th)

Prime decomposition worksheet (March 13th)