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Grobner Bases and Algebraic Geometry, Spring 2008

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

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.

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