Math 380, Spring 2018, Assignment 2

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

Read:

1. Section 1.2.
2. Section 1.3.

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

1. $\mathbb{V}(S)$ (the affine variety defined by some set $S$ of polynomials in $\mathsf{k}[x_1,\dots,x_n]$).
2. Rational expression.
3. Rational parametrization (of an affine variety). (The book defines this, starting in the second paragraph after Definition 1 on page 15.)

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

1. Bound on the number of roots of a univariate polynomial of degree $d$.
2. Theorem concerning whether two different polynomial expressions in $\mathsf{k}[x_1,\dots,x_n]$ can define the same function from $\mathsf{k}^n$ to $\mathsf{k}$, when $\mathsf{k}$ is a finite field.
3. Theorem concerning whether two different polynomial expressions in $\mathsf{k}[x_1,\dots,x_n]$ can define the same function from $\mathsf{k}^n$ to $\mathsf{k}$, when $\mathsf{k}$ is an infinite field.
4. Formula showing how to write the union of two varieties as a variety.
5. Formula showing how to write the intersection of some collection of varieties as a variety.
6. Formula showing how to write the empty set as a variety.

Solve the following problems:

1. Section 1.2, problems 1, 4, 8, and 11.
2. Section 1.3, problems 1 and 4.
3. Let $\mathsf{k}$ be any field. Show that a subset $U$ of $\mathsf{k}^1$ is a variety if and only if it is either a finite set, or all of $\mathsf{k}^1$. (Hint: in class we showed that finite sets are varieties, and $\mathsf{k}^1$ itself is the variety defined by the equation $0=0$. For the other direction, suppose $U$ is infinite, and let $f$ be any polynomial vanishing at every point of $U$. What polynomial is $f$?)
4. Show by example that the conclusion of the previous problem is generally false for subsets of $\mathsf{k}^n$ when $n>1$. (Hint: find a variety in $\mathbb{R}^2$ which has infinitely many points but is not all of $\mathbb{R}^2$.)

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