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.)

Questions:

1. I know we covered in class how to extend a partial solution to a total solution, but is there a separate definition we need to know for a total solution or should we just say that a total solution is a partial solution that does not make everything vanish (of course, the definition would be more formal). I also wanted to confirm for problem one that we needed to find a GB first and then eliminate one variable at a time. I tried to set up a lex GB and encountered some pretty nasty multivariable polynomial long division and have generated at least 6 terms (incuding the given) and have not finished computing S-pairs. --Robert.Moray (talk) 09:00, 16 April 2013 (EDT)

A total solution (for an ideal \(I\subset k[x_1,\dots,x_n]\)) is a point of the corresponding variety; this is a point of \(k^n\). A partial solution is a point of the variety of some elimination ideal; for the \(l\)th elimination ideal this will be a point of \(k^{n-l}\). -Steven.Jackson (talk) 15:46, 17 April 2013 (EDT)

Thank you! That makes so much sense now! --Robert.Moray (talk) 18:41, 17 April 2013 (EDT)

1. For problem 1, I'm also having a hard time getting a G.b. just in either X or Y. OTOH, just looking at the two equations, it's easy to see how one can solve for X and Y to get 4 solutions.--Matthew.Lehman (talk) 03:15, 17 April 2013 (EDT)

I'm getting a moderately tedious but not unmanageable calculation using Groebner bases and the elimination theorem. At one point I did make a certain disastrous mistake -- I forgot to write one of the intermediate dividends in lex order, leading to a big mess. So take care with this (and other small mistakes). -Steven.Jackson (talk) 15:46, 17 April 2013 (EDT)

1. Answer to question 1\[I\cap k[y] = \left\langle y^3-y\right\rangle\] and \(I\cap k[x] = \left\langle x^4 - 4x^2 + 3\right\rangle\). As for solutions: by eliminating \(x\) one sees immediately that \(y\) must be 0, 1, or -1. Substituting into the original system and solving the result for \(x\), we obtain a total of four solutions, namely \((x,y) = (\sqrt{3},0)\) or \((-\sqrt{3},0)\) or \((1,1)\) or \((-1,-1)\). -Steven.Jackson (talk) 21:25, 17 April 2013 (EDT)
2. After 10 calculations on 5 G.b.s, I got the ideal in Y twice, but none of them eliminated Y to get an ideal in X.--Matthew.Lehman (talk) 22:33, 17 April 2013 (EDT)

The usual lex order will always eliminate \(x\). In order to eliminate \(y\), you need to change the monomial order: use a new, lex-like order in which \(y\) is considered the leftmost (i.e. most significant) variable. -Steven.Jackson (talk) 22:48, 17 April 2013 (EDT)