Difference between revisions of "Algebraic variety"
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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 ''x<sup>2</sup> + y<sup>2</sup> +z<sup>2</sup> -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 ''x<sup>2</sup> + y<sup>2</sup> +z<sup>2</sup> -1 = 0''.]] | ||
An '''algebraic variety''' ''V'' over a [[ring (mathematics)|ring]] ''K'' is defined as the set of all points in ''K'' satisfying a system of polynomial equations with coefficients in ''K''. | An '''algebraic variety''' ''V'' over a [[ring (mathematics)|ring]] ''K'' is defined as the set of all points in ''K'' satisfying a system of polynomial equations with coefficients in ''K''. | ||
[[category:mathematics]] | [[category:mathematics]] | ||
Revision as of 18:10, May 29, 2007
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.
An algebraic variety V over a ring K is defined as the set of all points in K satisfying a system of polynomial equations with coefficients in K.