An example of an algebraic set is affine n-space itself.
Understanding Algebraic Sets
An algebraic set is a set of points that satisfy a system of polynomial equations. These sets are fundamental in algebraic geometry. Let's break this down further with examples based on the reference provided.
- Definition: An algebraic set is the set of solutions to one or more polynomial equations.
Examples of Algebraic Sets
According to the reference, here are some straightforward examples:
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Affine n-space (An): This is the entire n-dimensional space. It's an algebraic set because it's defined by the equation 0 = 0.
- Mathematically: An = Z(0). Z(0) represents the set of points where the polynomial 0 evaluates to zero, which is, of course, everywhere.
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The Empty Set (∅): The empty set is considered an algebraic set because it can be defined by the equation 1 = 0. There are no solutions to this equation.
- Mathematically: ∅ = Z(1). Z(1) represents the set of points where the polynomial 1 evaluates to zero, which is nowhere.
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A Single Point in An: Any single point (a1,...,an) in affine n-space is an algebraic set. This point is defined by the equations x1 − a1 = 0, x2 − a2 = 0, ..., xn − an = 0.
- Mathematically: (a1,...,an) = Z(x1 − a1,...,xn − an). This means the set containing only the point (a1,...,an) is the set of points where all the polynomials xi - ai are simultaneously zero.
Summarizing the Examples in a Table
| Algebraic Set | Description | Defining Equation(s) |
|---|---|---|
| Affine n-space (An) | The entire n-dimensional space. | 0 = 0 |
| Empty Set (∅) | A set containing no points. | 1 = 0 |
| Single Point (a1,...,an) | A set containing only one point. | xi - ai = 0 for all i |
