# Grobner Bases and Algebraic Geometry, Summer 2006

### 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, prime ideals, and prime(ary) decomposition of ideals.
- Quotient rings.
- Computing in quotient rings and 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.

The following Maple worksheets are from the course.

Groebner9.mws Maple 9 (classic) worksheet from Wednesday June 7 demo

Groebner10.mw Maple 10 (standard) worksheet from Wednesday June 7

resultant.mws worksheet from June 16

ressolve.mws worksheet from June 21

radical.mws worksheet from June 28

ideal intersection worksheet (Intersect.mws) from June 30

ideal quotient worksheet (IdQuo.mws) from July 5

prime decomposition examples (PrimeDecomp.mws) from July 14

scattering 3 (6) points in the square from August 2

minimal polynomials worksheet from August 2