Affine space
Affine algebraic geometry

In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame. Then, a polynomial function is a function such that the image of any point is the value of some multivariate polynomial function of the coordinates of the point. As a change of affine coordinates may be expressed by linear functions (more precisely affine functions) of the coordinates, this definition is independent of a particular choice of coordinates.

The choice of a system of affine coordinates for an affine space A k n {\displaystyle \mathbb {A} _{k}^{n}} of dimension n over a field k induces an affine isomorphism between A k n {\displaystyle \mathbb {A} _{k}^{n}} and the affine coordinate space k. This explains why, for simplification, many textbooks write A k n = k n {\displaystyle \mathbb {A} _{k}^{n}=k^{n}}, and introduce affine algebraic varieties as the common zeros of polynomial functions over k.

As the whole affine space is the set of the common zeros of the zero polynomial, affine spaces are affine algebraic varieties.
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