Introduction to Gröbner Bases
8 lectures given by Kyriakos Kalorkoti .Room JCMB 6206, Thursday 1400-1500.
Starting Thursday 19th October.
Last lecture Thursday, 7 December 2006.
Gröbner bases have become an increasingly used tool in various applications, essentially anything that can be modelled using polynomials might benefit from this tool. This short course will provide an introduction to this area focussing on a natural development of the idea and key algorithm.
Syllabus
Basic concepts from commutative algebra: rings, polynomial rings, ideals, factor rings, fields, algebraic closure. Links with algebraic geometry: varieties, Hilbert Basis Theorem and the Nullstellensatz. Ideal membership problem, attempts at developing an algorithm leading to concepts of admissible orders and Gröbner basis. Existence of Gröbner bases (non constructive proof), Buchberger algorithm. Applications, e.g., existence of solutions to simultaneous equations. Effect of admissible order used to compute a basis, both on information deduced (e.g., number of solutions to equations) and runtime. Brief look at complexity implications.Reading
D. Cox, J. Little and D. O'Shea. Ideals, Varieties and Algorithms, Second Edition, Springer, 1996.
J. von zur Gathen and J. Gerhard. Modern Computer Algebra, Second Edition, Cambridge University Press, 2003.
K. Kalorkoti. Introduction to Computer Algebra UG4 course lecture notes. (An appropriate extract will be supplied.)