---
tag: VHFMQQYSTN
kind: concept
name: Algebraic set
msc: [14A10, 14A05]
sources:
  - url: https://en.wikipedia.org/wiki/Algebraic_variety
---

## Summary

An algebraic set is the set of common zeros, in the affine space $k^n$ over a field $k$, of a family of polynomials.
The polynomials are in $n$ variables.
Algebraic sets are the basic geometric objects of classical algebraic geometry.
They are the closed sets of the Zariski topology on $k^n$.
Over an algebraically closed field they correspond to radical ideals of the polynomial ring.

## Explanations

### L1 human

Write down some polynomial equations in $x_1, \dots, x_n$ and look at all points that solve every one of them at once.
That set of solutions is an algebraic set.
In the plane, the equation $x^2 + y^2 - 1 = 0$ gives the unit circle, and the two equations $y = 0$ and $x = 0$ together give just the origin.
Lines, conics, finite sets of points and the whole plane are algebraic sets, but a disc or a half-line is not.