Exercise 109.31.1. Consider the affine curve $X$ given by the equation $t^2 = s^5 + 8$ in $\mathbf{C}^2$ with coordinates $s, t$. Let $x \in X$ be the point with coordinates $(1, 3)$. Let $U = X \setminus \{ x\} $. Prove that there is a regular function on $U$ which is not the restriction of a regular function on $\mathbf{C}^2$, i.e., is not the restriction of a polynomial in $s$ and $t$ to $U$.

## 109.31 Regular functions

Exercise 109.31.2. Let $n \geq 2$. Let $E \subset \mathbf{C}^ n$ be a finite subset. Show that any regular function on $\mathbf{C}^ n \setminus E$ is a polynomial.

Exercise 109.31.3. Let $X \subset \mathbf{C}^ n$ be an affine variety. Let us say $X$ is a *cone* if $x = (a_1, \ldots , a_ n) \in X$ and $\lambda \in \mathbf{C}$ implies $(\lambda a_1, \ldots , \lambda a_ n) \in X$. Of course, if $\mathfrak p \subset \mathbf{C}[x_1, \ldots , x_ n]$ is a prime ideal generated by homogeneous polynomials in $x_1, \ldots , x_ n$, then the affine variety $X = V(\mathfrak p) \subset \mathbf{C}^ n$ is a cone. Show that conversely the prime ideal $\mathfrak p \subset \mathbf{C}[x_1, \ldots , x_ n]$ corresponding to a cone can be generated by homogeneous polynomials in $x_1, \ldots , x_ n$.

Exercise 109.31.4. Give an example of an affine variety $X \subset \mathbf{C}^ n$ which is a cone (see Exercise 109.31.3) and a regular function $f$ on $U = X \setminus \{ (0, \ldots , 0)\} $ which is not the restriction of a polynomial function on $\mathbf{C}^ n$.

Exercise 109.31.5. In this exercise we try to see what happens with regular functions over non-algebraically closed fields. Let $k$ be a field. Let $Z \subset k^ n$ be a Zariski locally closed subset, i.e., there exist ideals $I \subset J \subset k[x_1, \ldots , x_ n]$ such that

A function $\varphi : Z \to k$ is said to be *regular* if for every $z \in Z$ there exists a Zariski open neighbourhood $z \in U \subset Z$ and polynomials $f, g \in k[x_1, \ldots , x_ n]$ such that $g(u) \not= 0$ for all $u \in U$ and such that $\varphi (u) = f(u)/g(u)$ for all $u \in U$.

If $k = \bar k$ and $Z = k^ n$ show that regular functions are given by polynomials. (Only do this if you haven't seen this argument before.)

If $k$ is finite show that (a) every function $\varphi $ is regular, (b) the ring of regular functions is finite dimensional over $k$. (If you like you can take $Z = k^ n$ and even $n = 1$.)

If $k = \mathbf{R}$ give an example of a regular function on $Z = \mathbf{R}$ which is not given by a polynomial.

If $k = \mathbf{Q}_ p$ give an example of a regular function on $Z = \mathbf{Q}_ p$ which is not given by a polynomial.

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