Dimension of an algebraic variety

In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V .

For example, an algebraic curve has by definition dimension 1. That means that any two rational functions F and G on it must satisfy some polynomial relation

: P ( F , G ) = 0.

That implies that F and G are constrained to take related values (up to some finite freedom of choice): they cannot be truly independent.

== Formal definition ==

For an algebraic variety V over a field K , the dimension of V is the transcendence degree over K of the function field K(V) of all rational functions on V , with values in K .

For the function field even to be defined, V here must be an irreducible algebraic set; in which case the function field (for an affine variety) is just the field of fractions of the co-ordinate ring of V . It is easy to define by polynomials sets that have mixed dimension : a union of a curve and a plane in space, for example. These fail to be irreducible.