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).

Questions:

1. With regard to defining the leading term ideal of a given ideal, wouldn't I need to define a monomial ordering so that I could identify the leading term, or is it sufficient to define an ideal and then the leading term ideal by just saying it is an ideal generated by the leading terms of the generators of the defined ideal? --Robert.Moray (talk) 19:49, 12 April 2013 (EDT)

Probably the most careful definitions will start with something like "Fix a monomial ordering \(\leq\) on the polynomial ring \(k[x_1,\dots,x_n]\). For any ideal \(I\) in this ring, define \(LT(I)\), the leading term ideal of \(I\), to be..." -Steven.Jackson (talk) 20:03, 12 April 2013 (EDT)