CECM Colloquium

**Wednesday April 11, 2001
in K9509, SFU
**

3:30 - 4:20

Dr Michael Monagan, SFU

Talks on "Buchberger's algorithm for Groebner bases."

**Abstract:**
Let k be a field and F = {f1,...,fn} be a basis for an ideal in
k[x1,...,xn].
If F is linear, questions such as the dimension of the solution set of
F 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.