An algebraic set is the locus of zeros of a collection of polynomials. For example, the circle is the set of zeros of 
and the point at 
is the set of zeros of 
and 
. The algebraic set 
is the set of solutions to 
. 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, 
and 
intersect at 
, i.e., where 
and 
. 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 
is the set of solutions to 
and 
is the set of solutions to 
, then 
is the set of solutions to 
. Consequently, the algebraic sets are the closed sets
in a topology, called the Zariski
topology.
The set of polynomials vanishing on an algebraic set 
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.
