2005/05/02 | 仿射簇[Affine Variety]
类别(∑〖数学〗) | 评论(0) | 阅读(85) | 发表于 13:07
An affine variety V is a variety contained in affine space. For example,
(1)

is the cone, and
(2)

is a conic section, which is a subvariety of the cone. The cone can be written to indicate that it is the variety corresponding to Naturally, many other polynomials vanish on , in fact all polynomials in .The set I(C) is an ideal in the polynomial ring . Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by and .
A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map is a morphism from to . Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety is isomorphic to the cone via the coordinate change .
Many polynomials f may be factored, for instance ,and then . Consequently, only [u]irreducible polynomials ,and more generally only prime ideals are used in the definition of a variety. An affine variety V is the set of common zeros of a collection of polynomials , ..., , i.e.,
(3)

as long as the ideal is a prime ideal. More classically, an affine variety is defined by any set of polynomials, i.e., what is now called an algebraic set. Most points in V will have dimension , but V may have singular points like the origin in the cone.
When V is one-dimensional generically (at almost all points), which typically occurs when , then V is called a curve. When V is two-dimensional, it is called a surface. In the case of complex affine space, a curve is a Riemann surface, possibly with some singularities.

Mathematica has a built-in function ImplicitPlot in the Mathematica add-on package Graphics`ImplicitPlot` (which can be loaded with the command <<Graphics`) that will graph affine varieties in the real affine plane. For example, the following graphs a hyperbola and a circle.

<<Graphics`;
Show[GraphicsArray[{
ImplicitPlot[x^2 - y^2 == 1, {x, -2, 2}, DisplayFunction -> Identity],
ImplicitPlot[x^2 + y^2 == 1, {x, -2, 2}, DisplayFunction -> Identity]
}]]
An extension to this function called ImplicitPlot3D (Wilkinson) that can be used to plot affine varieties in three-dimensional space can be downloaded from the Mathematica Information Center.
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