An algebraic set is the locus of zeros of a collection of polynomials. For example, the circle is the set of zeros of x^2 + y^2 - 1 and the point at (0, 0) is the set of zeros of x and y. The algebraic set {(x, 0)} union {(0, y)} is the set of solutions to x y = 0. It decomposes into two irreducible algebraic sets, called algebraic varieties. In general, an algebraic set can be written uniquely as the finite union of algebraic varieties.
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