Math 480, Spring 2013, Assignment 6

Carefully define the following terms, then give one example and one non-example of each:

1. Leading term ideal (of a given ideal).
2. Groebner basis (of a given ideal with respect to a given monomial order).

Carefully state the following theorems (you need not prove them):

1. Hilbert Basis Theorem.
2. Theorem on ascending chains of ideals (in polynomial rings).
3. Theorem on uniqueness of remainders with respect to Groebner bases.

Carefully state how to execute the following algorithms:

1. Ideal membership algorithm (given a precomputed Groebner basis).