#### Assignments

#### Solutions

The solutions are large.

Assignment 1

Assignment 2

Assignment 3

Assignment 4

Assignment 5

Assignment 6

# Commutative Algebra and Algebraic Geometry

### MATH 441 and MATH 819, Spring 2016

## Content

- Varieties and ideals.
- Monomial orderings and the division algorithm.
- Dickson's lemma and the Hilbert basis theorem.
- Gröbner bases and Buchberger's algorithm.
- Solving systems of polynomial equations using Gröbner bases.
- Implicitization.
- Resultants.
- Hilbert's Nullstellensatz and radical ideals.
- Irreducible varieties and prime ideals.
- Decomposition of ideals and varieties.
- Quotient rings; construction, computation and application.
- Proving theorems in geometry.

#### Textbook

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

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

#### Software

We will use Maple extensively for calculations and programming in this course. The university has a site license for Maple. Maple is installed on the PCs and MACs in the assignment lab, the CECM lab, university open labs and the library. If you haven't used Maple before, the following Maple worksheet will get you started: Getting Started with Maple (.mw). You must first save it on your desktop, then open it from inside Maple and execute the examples to see what Maple does. Here is a .pdf version of the same worksheet so that you can see what it should look like Getting Started .pdf version.

## Handouts

course information sheet (.txt)

The Maple appendix at the back of the textbook is out of date. David Cox sent me this updated version: NewMapleAppendix. It contains information about Maple's Groebner package (for Chapters 2 and 3) and Maple's PolynomialIdeals package (Chapter 4). I have also put together a Maple demo worksheet containing examples showing you how to use the Groebner package below.

#### Maple Worksheets

Intro.mw (demo from first day) Intro.pdf (.pdf version)

GroebnerDemo.mw Examples for using the Groebner package and PolynomialIdeals package (.mw).

GroebnerDemo.pdf Examples for using the Groebner package and PolynomialIdeals package (.pdf).

DivAlg.mw or DivAlg.pdf The divison algorithm

Intersect.mw and Intersect.pdf Ideal intersection examples from Tuesday March 8th

IdealQuotient.mw or IdealQuotient.pdf Ideal quotient examples from Thursday March 10th

Primality.mw or Primality.pdf) Testing ideals for primality from Tuesday March 15th

PrimeDecomp.mw or PrimeDecomp.pdf Prime decomposition examples from Thursday March 17th

QuoRings.mws or QuoRings.pdf Quotient ring examples from Thursday March 24th

AutoGeo.mw or AutoGeo.pdf Parallelogram theorem worksheet from Tuesday March 29th

Scattering.mw or Scattering.pdf Scattering points in the unit square (from Thursday March 31st)