# Grobner Bases and Algebraic Geometry

### MATH 439, MATH 739, MATH 819, Spring 2010

### 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

We will cover Chapters 1,2,4 and selected topics from Chapters 3,5,6.

#### 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

Maple worksheet showing how to use Maple (.mws)

Course info sheet (.txt)

Maple worksheet demo first day (January 4th) (.mws)

Maple worksheet for how to use Groebner package (.mws)

Maple worksheet on implicitization (February 10th) (.mw)

Maple worksheet for Groebner[Basis] for the Trinks system (February 10th)

Resultant worksheet (February 21st) (.mws)

Maple worksheet on the radical of an ideal (March 4th) (.mws)

Ideal intersection worksheet (March 11th) (.mws)

Ideal quotient worksheet (March 13th) (.mws)

Prime decomposition worksheet (March 20th) (.mws)

Quotient rings example worksheet (.mws) from April 1

Midpoint on Parallelogram Theorem worksheet (April 7) (.mws)

scattering 3 (6) points in the square from April 12 (.mws)

minimal polynomials worksheet from April 12 (.mw)

Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility

ISSAC '08 paper (.pdf) by De Loera, Lee, Malkin, and Margulies.

Maple worksheet (.mw) by Michael Monagan