Math 480, Spring 2013, Assignment 8

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

1. Elimination ideal.
2. Partial solution.
3. Total solution.

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

1. Elimination theorem.
2. Extension theorem.

Do the following problems:

1. Consider the system of equations \(x^2+2y^2=3, x^2+xy+y^2 = 3\). Let \(I\) be the ideal generated by these equations. Find bases for \(I\cap k[x]\) and \(I\cap k[y]\). Then find all solutions of these equations.
2. Let us say that a monomial ordering on \(k[x_1,\dots,x_n]\) is an elimination order with respect to the first \(l\) variables if each monomial involving any of the first \(l\) variables is greater than any monomial not involving the first \(l\) variables. Formulate and prove a version of the Elimination Theorem in which lex order is replaced by an arbitrary elimination order.
3. Give an example of an elimination order other than lex. (Hint: the Wikipedia article on monomial orders is very helpful, especially the "Related notions" section.)