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[[Image:Sphere-wireframe.png|right|thumb|200px|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''.]]

[[Image:Sphere-wireframe.png|right|thumb|200px|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 (mathematics)|ring]] ''K'' is defined as the set of all points in <math>K^n</math> satisfying a system of [[polynomial]] equations with [[coefficient]]s in ''K''.

Formally, an '''algebraic variety''' ''V'' over a [[ring (mathematics)|ring]] ''K'' is defined as the set of all points in <math>K^n</math> satisfying a system of [[polynomial]] equations with [[coefficient]]s in ''K''.

* The union of two intersecting lines is a variety.  For example, the set consisting of <math>xy=0</math> is the x-axis together with the y-axis.

* The union of two intersecting lines is a variety.  For example, the set consisting of <math>xy=0</math> 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 section|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!

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 section]]s (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!

[[category:mathematics]]

[[Category:Mathematics]]