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Commutative Algebra and Algebraic Geometry

MATH 441, MATH 741, MATH 819, Spring 2012

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. 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 (.mws)

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The Maple appendix at the back of the textbook is out of date.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 (Chapters 2 and 3) and Maple's PolynomialIdeals package (Chapter 4). I will put together a Maple demo worksheet containing examples showing you how to use the Groebner package and PolynomialIdeals package.

Maple Worksheets

GroebnerDemo.mws Examples for using the Groebner package from Tuesday February 7th
DivAlg.mw or DivAlg.mws The divison algorithm (Thursday February 9th)
Trinks.mw or Trinks.mws on Groebner basis for Trinks system
Implicit.mw or Implicit.mws on implicitization examples
resultant.mws resultant examples (February 24th)
radical.mws radical examples (March 6th)
Intersect.mws ideal intersection examples (March 8th)
IdealQuotient.mws ideal quotient examples (March 13th)
PrimeDecomp.mws prime decomposition examples (March 20th)
QuoRings.mws quotient ring examples (March 27th)
AutoGeo.mws midpoint on parallelogram theorem worksheet (March 28th)
Scattering.mw scattering points in the unit square (April 3rd)
Minpoly.mw minimal polynomials worksheet (April 3rd)

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

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