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Grobner Bases and Algebraic Geometry

MATH 439, MATH 739, MATH 819, Spring 2010


  • 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.


Ideals, Varieties and Algorithms by Cox, Little, O'Shea
We will cover Chapters 1,2,4 and selected topics from Chapters 3,5,6.


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.

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