Affine algebraic sets,

Question

Which of the following sets are algebraic?

$\{(x,x^2)|x\in\mathbb{R}\}\subseteq\mathbb{R}^2$

$\{(x,x^2, x^3)|x\in\mathbb{R}\}\subseteq\mathbb{R}^3$

$\{(x,x^{-1})|x\in\mathbb{R}\setminus\{0\}\}\subseteq\mathbb{R}^2$

A set $X\subseteq\mathbb{A}^n$ is algebraic, if $X=V(S)$ for $S\subseteq k[X_1,\dotso, X_n]$ where $k$ is an algebraic closed field.

That means I have to find a set of polynomials, which describes the given set.

It is $V(Y-X^2)=\{(x,x^2)|x\in\mathbb{R}\}$

It is $V(Y-X^2, Z-X^3)=\{(x,x^2,x^3)|x\in\mathbb{R}\}$

It is $V(XY-1)=\{(x,x^{-1})|x\in\mathbb{R}\setminus\{0\}\}$

Every set is algebraic.

Am I right? Thanks in advance.

Answer 1

In the context of this question, the field $k$ is $\mathbb{R}$, which of course, is not algebraically closed.

Thus, for this question,

• $\mathbb{A}^n$ refers to $\mathbb{A}^n(\mathbb{R})$.$\\[4pt]$
• $X\subseteq\mathbb{A}^n$ is algebraic if $X=V(S)$ for some $S\subseteq \mathbb{R}[x_1,...,x_n]$.

Allowing that minor correction, your answers are fine.