Math 480, Spring 2013, Assignment 2

Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.

- F. Klein

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

1. Affine variety (defined by the polynomials \(f_1,\dots,f_s\)).
2. Rational function (of \(t_1,\dots,t_m\)). (Note that it would be better to call these objects rational expressions. Why?)
3. Polynomial parametrization (of an affine variety).
4. Rational parametrization (of an affine variety).
5. Implicit representation (of an affine variety).
6. Implicitization problem.

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

1. Theorem on unions and intersections of affine varieties (Lemma 2 in section 1.2).

Solve the following problems:

1. Working in \(\mathbb{R}^2\), sketch the variety \(\mathbf{V}(x^2-y^2)\).
2. Working in \(\mathbb{R}^3\), sketch the variety \(\mathbf{V}(xz^2-xy)\).
3. Working in \(\mathbb{R}^3\), sketch the variety \(\mathbf{V}(x, z^2-y)\).
4. Working in \(\mathbb{R}^3\), sketch the variety \(\mathbf{V}(x-1, y-2, z-3)\).
5. Show that any single-point set \(\{(a_1,\dots,a_n)\}\subset k^n\) is an affine variety.
6. Show that any finite subset of \(k^n\) is an affine variety. (Hint: use the previous problem together with the theorem on unions and intersections of affine varieties.)
7. Show that \(\mathbb{Z}\subset\mathbb{R}^1\) is not an affine variety. (Hint: suppose to the contrary that \(\mathbb{Z}\) is the variety defined by the polynomials \(f_1,\dots,f_s\). Then each \(f_i\) is a polynomial that vanishes at every integer. What can you conclude about \(f_i\)?) In light of the previous problem, why doesn't this contradict the theorem on unions and intersections?
8. Find an implicit representation of the affine variety parametrized by \(x = \frac{t}{1+t}, y = 1 - \frac{1}{t^2}\). Does the parametrization cover every point of this variety?

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Questions:

• I am slightly confused over the notion of a non-example of an affine variety. Would this be the affine variety of a polynomial with no roots? - --Rob Moray

Well, non-example isn't exactly a precise term; I suppose one could argue that a chair is a non-example of an affine variety. But the spirit of the question is to give a subset of \(k^n\) which is not an affine variety, i.e. not definable by any system of polynomial equations -- see problem 7. (The empty set is not a non-example since it is the variety \(\mathbf{V}(\{1\})\).) - Steven.Jackson (talk) 21:25, 5 February 2013 (EST)