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.