Math 480, Spring 2013, Assignment 9

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

1. Projection map.
2. Zariski closure.
3. Polynomial map.
4. Rational map.
5. Graph of a map.

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

1. Geometric extension theorem.
2. Closure theorem.
3. Polynomial implicitization theorem (Theorem 3.3.1).
4. Rational implicitization theorem (Theorem 3.3.2).

Do the following problems:

1. Let \(S\) be the image of the polynomial map given by \((x, y, z) = (uv, uv^2, u^2)\). Find the ideal of the Zariski closure of \(S\). Then (working over \(\mathbb{C}\)) find all the points of the Zariski closure of \(S\) which do not lie in \(S\) itself.

Questions:

1. I am a little bit confused by the notation used for the problem. So I assume we have a polynomial map here and it is taking a subvariety to another subvariety, and it looks like there are three variables in the preimage and two variables in the image. The problem is that, for whatever reason, I am having a problem translating the concept from the class notes to the homework problem. For instance, with the twisted cubic, we introduced (x,y,z) to make a variety, but we took one variable to more variables, not three variables to less variables. In the twisted cubic, we introduced three variables which led to a variety whose terms we could use Buchberger's algorithm on to obtain a lexicographic G.B with respect to the three variables only. Is this the same principle that would be applied here? If so, should I assume I should find a way of translating between (x,y,z) and (u,v)? --Robert.Moray (talk) 16:03, 15 April 2013 (EDT)

The map is meant to take (all of) \(k^2\) into \(k^3\). The input variables are \(u\) and \(v\) while the output variables are \(x, y,\) and \(z\). Hope that helps. -Steven.Jackson (talk) 16:19, 15 April 2013 (EDT)

1. Please remind me again, will the test include material through this assignment?--Matthew.Lehman (talk) 13:41, 17 April 2013 (EDT)

Yes. -Steven.Jackson (talk) 15:48, 17 April 2013 (EDT)

1. On this homework problem I have been doing Buchberger's algorithm and keep generating terms in the grobner basis. I do not know if the algorithm is supposed to keep iterating like this. Did anyone get anything differently or am I generating the ideal incorrectly? I have an ideal \(I=<u^2-z,uv^2-y.uv-x> \) to start and am using a lexicographic order where u > v > x > y > z. --Robert.Moray (talk) 18:38, 17 April 2013 (EDT)