Algebraic set
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In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K) of a set of simultaneous equations
- P1(X1, ...,Xn) = 0
- P2(X1, ...,Xn) = 0
and so on up to
- Pm(X1, ...,Xn) = 0
for some integer m. That is, we consider the simultaneous solution set of these equations applied to vectors
- (x1, ...,xn)
with the xi taken from K.
Algebraic sets are the primitive objects of algebraic geometry. To get the standard concept of algebraic variety, however, three extra aspects need to be introduced:
- K should be an algebraically closed field, for example the complex numbers
- The irreducible sets are the fundamental objects
- Homogeneous polynomials and projective space should be introduced, to cut down exceptional cases such as parallel lines.
For example, if K is the real number field, an algebraic set can easily be the empty set in cases where the complex number solutions are numerous. Under the first two conditions there is a satisfactory definition of dimension.
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