Algebraic Set

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 xy=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.

The intersection of two algebraic sets is an algebraic set corresponding to the union of the polynomials. For example, x=0 and y=0 intersect at (0,0), i.e., where x=0 and y=0. In fact, the intersection of an arbitrary number of algebraic sets is itself an algebraic set. However, only a finite union of algebraic sets is algebraic. If X is the set of solutions to f_i=0 and Y is the set of solutions to g_j=0, then X union Y is the set of solutions to f_ig_j=0. Consequently, the algebraic sets are the closed sets in a topology, called the Zariski topology.

The set of polynomials vanishing on an algebraic set X is an ideal in the polynomial ring. Conversely, any ideal defines an algebraic set since it is a collection of polynomials. Hilbert's Nullstellensatz describes the precise relationship between ideals and algebraic sets.

See also

Algebraic Variety, Category Theory, Commutative Algebra, Conic Section, Hilbert's Nullstellensatz, Ideal, Prime Ideal, Scheme, Zariski Topology

This entry contributed by Todd Rowland

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Rowland, Todd. "Algebraic Set." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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