an algebraic variety

If a single polynomial equation cuts out a curve in the plane, what do you get from several equations in many variables at once? An algebraic variety is the answer: it is the set of points in space (of any number of dimensions) that satisfy a whole system of polynomial equations simultaneously. A line, a circle, a sphere (x^2 + y^2 + z^2 = 1), the surface where two equations both hold — each is a variety. It is the higher-dimensional, many-equation generalization of the algebraic curve, and it is the central object of algebraic geometry.

Concretely, pick polynomials f_1, f_2, ..., f_k in variables x_1, ..., x_n. The variety they define is every point (x_1, ..., x_n) for which f_1 = 0 AND f_2 = 0 AND ... AND f_k = 0 — the common zero set, the intersection of each equation's solution set. One equation in 3D typically gives a surface (a 2-dimensional variety); add a second independent equation and the solutions usually drop to a curve (1-dimensional); a third can pin it down to isolated points (dimension 0). So the DIMENSION of a variety is, roughly, the number of free coordinates left after the equations have done their constraining. As with curves, a variety is IRREDUCIBLE if it is not the union of two smaller varieties, and the irreducible ones are the genuine building blocks.

Varieties are where geometry and algebra fuse: every geometric question (Do these shapes meet? How? With what multiplicity?) becomes an algebra question about the polynomials, and vice versa — a dictionary made precise in the twentieth century by ideals, rings, and eventually schemes. This powers everything from robotics motion-planning (the configurations of a robot arm form a variety) to physics and coding theory. One honest subtlety: 'variety' is used slightly differently by different authors — some demand irreducibility, some work only over the complex numbers, some allow the more general 'scheme'. The shared core, though, is always the same: solutions of polynomial systems, viewed geometrically.