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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The Maple appendix at the back of the textbook is out of date.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)

p. (778) 782-4279 · f. (778) 782-4947 · Shrum Science K 10501 · Department of Mathematics · 8888 University Drive · Burnaby · BC · V5A 1S6 · Canada