Visualizing affine algebraic varieties

An affine algebraic variety is an irreducible closed subset of {A_k}^n (with the induced topology) as discussed in the previous two posts.

A non-constant polynomial p(x,y) in the variables x and y defines a plane curve affine variety Z(p) \subset {A_k}^2. Below is the variety for the plane elliptic curve p(x,y) := y^2-x^3+x .

Varieties 1

The set of four points \{ (-2,-1),(-1,1),(1,-1),(1,2) \} in {A_k}^2 is an affine variety. It is the intersection of an ellipse Z(x^2+y^2-xy-3) and a hyperbola Z(3x^2-y^2-xy+2x+2y-3).

Varieties 2

A quadratic is an affine algebraic variety defined by a single quadratic polynomial. In {A_k}^2 these are the plane conics; circles, ellipses, parabolas, and hyperbolas in \mathbb{R}^2. In {A_k}^3 these are; spheres, ellipsoids, paraboloids, and hyperboloids in \mathbb{R}^3.

Below are examples of the two paraboloid affine algebraic varieties, a hyperbolic paraboloid (a Pringle has this surface!) Z(xy+z) and an elliptic paraboloid Z(x^2+y^2-z).

Varieties 3 (Pringle)

Varieties 4

Below are examples of two hyperboloid affine algebraic varieties, a hyperboloid of one sheet Z(x^2-x+y^2+yz) and a hyperboliod of two sheets Z(x^2+y^2-z^2+1).

Varieties 5

Varieties 6

Plane curves and quadratics defined by a single polynomial are called hyper-surfaces. Some affine algebraic varieties are intersections of hyper-surfaces.

Below is an example of an affine algebraic variety, the intersection of two quadratics Z(x^2+y^2+z^2-1) and Z(x^2-x+y^2) that is a `figure of eight’ curve in \mathbb{R}^3.

Varieties 7

Irreducible varieties such as the plane curve Z(y^2-x^3+x) in {A_\mathbb{R}}^2 and the surface Z((x^2-y^2)^2-2x^2-2y^2-16z^2+1) in {A_\mathbb{R}}^3 are irreducible hyper-surfaces. These are visualised below; the first has components that are connected in Euclidean topology, the second has five components that meet at four singular points.

Varieties 8

Varieties 9

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One Response to Visualizing affine algebraic varieties

  1. Pingback: Finding real roots of polynomial systems using Macaulay 2 | kevinjdavis2013

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