Groebner bases, Buchberger's algorithm, and applications

Michael Monagan

Let k be a field and F = {f1,...,fn} be a basis for an ideal in k[x1,...,xn].
If f1,...,fn are linear, questions such as the dimension of the solution set of
and whether f is in span(F) can be readily answered by first reducing F to row
Echelon form using Gaussian elimination.  That's linear algebra.
What happens when the polynomials in F are non-linear?
In 1965 Buchberger defined Groebner bases and gave an algorithm for computing
a Groebner basis for the ideal generated by F, and in so doing completely
generalized linear algebra to polynomial systems.  Two important applications
of this theory are a method to effectively solve a system of polynomial
equations and a way to compute standard representatives for the equivalence
classes of the quotient ring R = k[x1,...,xn]/I thus allowing effective
computation in R.  Another result is an effective algorithm for computing the
prime decomposition of the radical of an ideal in k[x1,...,xn].
This constructive theory has revolutionized the field of algebraic geometry
and the scope of its applications is now being determined.

In this talk I will explain how and why Buchberger's algorithm works.
I will summarize the main known theoretical results about Groebner
bases and show some lovely applications from different areas of mathematics.
No a priori knowledge of Groebner bases is assumed or expected.