Spring 2002, MWF 12:30-1:20
1. Ideals, Varieties, and Algorithms, Cox, Little and O'Shea, Springer-Verlag, UTM, Second Edition, New York 1996
2. Using Algebraic Geometry,
Cox, Little and O'Shea, Springer-Verlag, GTM, New York 1998.
Other suggested references
An Introduction to Gröbner
Bases, Adams and Loustaunau, AMS 1994. Computational Methods in Commutative Algebra
and Algebraic Geometry, Vasconceles,
Springer 1997. Applications of Computational Algebraic Geometry, Cox and
Sturmfels editors, AMS 1998.
Prerequisites
Familiarity with basic commutative
algebra and some interest in computations.
This course is intended to be a
first year graduate level introduction to the theory of Gröbner bases - a
cornerstone of computational algebra
and computational algebraic geometry.
Gröbner bases provide solution techniques for systems of polynomial equations and hence play a
central role in problems that are modeled by polynomial equations -- like robotics, computer aided geometric
design and coding theory. The ability to compute examples in algebra and
algebraic geometry have also led to several new results and theories in these
fields.
The course will be in two parts. The first part will cover the essentials of the theory of Gröbner bases - more specifically chapters 2-5 and 9 of Cox, Little and O'Shea's first book "Ideals, Varieties and Algorithms". The second part will focus on some of the applications of Gröbner bases theory as described in the second book "Using Algebraic Goemetry" by Cox, Little and O'Shea. Some of the specific topics to be covered are : solving polynomial equations, resultants, computations in local rings, free resolutions, and connections to polytopes. In the last two weeks of the course we will focus on specific applications of the theory to areas such as integer programming, coding theory, differential algebra and splines -- to name a few. These topics might be assigned as projects. There will be some emphasis on actual computations using the computer algebra package Macaulay 2.