Algebraic variety

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The unit sphere is an algebraic variety over the real numbers because it is the set of all roots of the polynomial equation x2 + y2 +z2 -1 = 0.

Formally, an algebraic variety V over a ring K is defined as the set of all points in Kn satisfying a system of polynomial equations with coefficients in K.

This has an easy intuitive definition: a variety is just the set of points where some polynomial(s) in several variables vanishes. Many familiar spaces are examples of algebraic varieties:

  • The parabola, i.e., the graph of y = x2, is a variety: it's the set of points in the plane \mathbb R^2 satisfying the polynomial relation yx2 = 0.
  • A circle is a variety too: it's the solution of x2 + y2 − 1 = 0 in the plane.
  • The union of two intersecting lines is a variety. For example, the set consisting of xy = 0 is the x-axis together with the y-axis.

Varieties are one of the primary objects of study of algebraic geometry. It is useful to group all of these objects together instead of studying each one separately, because many theorems can be proved about all varieties at once. For example, you may know that two conic sections (for example, two ellipses) generally intersect at four distinct points. You can also draw a graph and be convinced that the graphs of two cubics intersect at 9 distinct points. Bezout's theorem, a result about algebraic varieties in general, gives a way to compute the number of intersections of two varieties in terms of the degree of the polynomials of which they are zero sets. So both of these familiar facts are just cases of Bezout's theorem!

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